I know both of the Rambam and Ramban listed all 613 mitzvos, and they differ in some (I think like 16ish?)
What are the ones upon which they differ? If possible please include why they differ in specific instances.
Answer
The following is just my understanding of the sources, and if there are any mistakes (ex: in the Ramban's reasons to disagree with the Rambam, the titles given to mitzvos, etc.), please point them out to me. I used the Mishneh LaMelech's Derech Mitzvosecha § 1, this article, and the Ramban's summary at the end of his Hasagos to help with this list.
In summary: The Ramban removes 56 mitzvos (32 + 24) , and adds 63 mitzvos (32 + 31, see fn. 4).
If you would like to see the list of mitzvos in a more visually appealing format, click here to see a (very) Alpha version of a website I'm working on (Desktop only).
(Lettered entries aren't removed from Ramban's list, rather split in two)
The Ramban disagrees with the following positive mitzvos:
1) [#5] מצות תפילה - Prayer. Because it is rabbinic.
2) [#7] מצות על כל הצריך לישבע שישבע בשם השם - Anyone who has to swear to only swear in the name of Hashem. Because the verse is giving permisison, not commanding.
a) [#10] מצות קרית שמע שחרית וערבית - Reciting the Shema morning and evening. The Ramban splits this into two mitzvos, the other is below, addition #3.
b) [#28] מצות הקטרת קטרת - Burning the incense. The Ramban splits this into two mitzvos, the other is below, addition #4.
c) [#39] מצות תמידין בכל יום - The daily Tamid offering. The Ramban splits this into two mitzvos, the other is below, addition #5.
3) [#63] מצות מעשה העולה - The Olah offering procedure. Because as per Shoresh #12, this is just a detail of the mitzvah to bring offerings.
4) [#64] מצות מעשה החטאת - The Chatas offering procedure. Because of #3.
5) [#65] מצות מעשה האשם - The Asham offering procedure. Because of #3.
6) [#66] מצות מעשה זבח השלמים - The Shelamim offering procedure. Because of #3.
7) [#67] מצות קרבן מנחה - The Mincha offering. Because of #3.
8) [#85] מצות להטפל בהבאת הקרבן מחוצה לארץ לבית הבחירה - To take care of the offerings as they are brought from outside the land to the Temple. Because anyways we need to bring offerings to the Temple, regardless of where they are, so this isn't a new command.
d) [#94] מצות קיום מוצא שפתיו - To fulfill the vows of our mouths. The Ramban splits this mitzvah into two, this one being just about vows to do a mitzvah. The other is below, addition #9.
9) [#95] מצות דין הפרת נדרים - Nullification of vows. Because it is really a shelilah, to do all that which leaves our mouths and not to profane our words except through the annulment of Beis Din, a father, or a husband (see Shoresh #8).
10) [#96] מצות עניין טומאת נבילה - The concept of tumah of a neveilah. Because there is no mitzvah here, just a concept.
11) [#97] מצות טומאת שמנה שרצים - The concept of tumah of the eight sheratzim. Because of #10.
12) [#98] מצות עניין טומאת אוכלים - The concept of tumah of food. Because of #10.
13) [#99] מצות עניין טומאת נדה שטמאה ומטמאה - The concept of tumah of a niddah. Because of #10.
14) [#100] מצות עניין טומאת יולדת - The concept of tumah of a woman who gave birth. Because of #10.
15) [#101] מצות עניין טומאת מצורע - The concept of tumah of a metzora. Because of #10.
16) [#102] מצות עניין נגעי בגדים - The concept of tumah of tzaraas on a garment. Because of #10.
17) [#103] מצות עניין טומאת בית שיהיה בו נגע - The concept of tumah of tzaraas on a building. Because of #10.
18) [#104] מצות עניין טומאת זב להיות טמא ומטמא - The concept of tumah of a zav. Because of #10.
19) [#105] מצות עניין טומאת שכבת זרע שהוא טמא ומטמא - The concept of tumah of someone who had a seminal emission. Because of #10.
20) [#106] מצות עניין טומאת זבה שטמאה ומטמאה - The concept of tumah of a zavah. Because of #10.
21) [#107] מצות טומאה של מת - The concept of tumah of a corpse. Because of #10.
22) [#108] מצות מי נדה שמטמאין אדם טהור ומטהרין אדם טמא מטומאת מת בלבד - The concept of the water of a red heifer which purifies and contaminates. Because of #10.
e) [#126] מצות הפרשת תרומה גדולה - To separate Terumah Gedolah. The Ramban splits this mitzvah into two, this one being about separating Terumah. The other is below, addition #23.
f) [#127] מצות מעשר ראשון - To separate Maaser Rishon. The Ramban splits this mitzvah into two, this one being about separating Maaser Rishon. The other is below, addition #24.
g) [#130] מצות מעשר עני תחת מעשר שני בשנה שלישית - To separate Maaser Ani instead of Maaser Sheni in the third year. The Ramban splits this mitzvah into two, this one being about separating Maaser Ani. The other is below addition #25.
h) [#133] מצות חלה - To separate Challah. The Ramban splits this mitzvah into two, this one being about separating Chalah. The other is below, addition #26.
23) [#142] מצות לנגוש את הנכרי - To pressure a non Jew to repay a debt. Because this is a lav habah miklal aseh and since there is already a lav counted, this shouldn't be as well.
24) [#149] מצות בדיקת סימני בהמה וחיה - To check the kosher signs of domesticated and wild animals. Because this is a lav habah miklal aseh and since there is already a lav counted, this shouldn't be as well.
25) [#150] מצות בדיקת סימני העוף - To check the kosher signs of birds. Because of #24.
26) [#151] מצות בדיקת סימני חגבים - To check the kosher signs of grasshoppers. Because of #24.
27) [#152] מצות בדיקת סימני דגים - To check the kosher signs of fish. Because of #24.
i) [#153] מצות קדוש החודש - To sanctify the new month. The Ramban splits this mitzvah into two, this one being about establishing Rosh Chodesh. The other is below, addition #27.
28) [#198] מצות הלואה לנכרי ברבית אם יצטרך ללוות מה שאינו כן בישראל - To lend to a non Jew with interest if he needs a loan, unlike to a Jew. Because of #23.
29) [#226] מצות בית דין להרוג בסייף המחוייב - For Beis Din to kill those liable to decapitation. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
30) [#227] מצות בית דין להרוג בחנק המחוייב - For Beis Din to kill those liable to strangulation. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
31) [#228] מצות שישרפו מי שיתחייב שריפה - For Beis Din to kill those liable to death by fire. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
32) [#229] מצות על בית דין לסקול באבנים מי שיתחייב סקילה - For Beis Din to kill those liable to death by stoning. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
j) [#235] מצות עבודה בעבד כנעני לעולם - To rule over a Caananite slave forever. The Ramban splits this mitzvah into two, this one being to rule over them forever. The other is below, addition #31.
k) [#248]1 מצות דיני נחלות - The laws of inheritance. The Ramban splits this mitzvah into two, this one being to follow the laws of inheritance. The other is below, addition #32.
The Ramban adds the following positive mitzvos:
1) מצות אכילת מעשר שני ובכורות בירושלים - To Eat Maaser Sheni and Bechoros in Yerushalayim (source, description).
2) מצות אכילת תרומה כשהיא טהורה ולא בטומאתה - To eat Terumah when it is pure and not while it is impure (source, description).
3) מצות קרית שמע בערב - Reciting the Shema in the evening (source, description).
4) מצות הקטרת באין הארבים - Burning the afternoon incense (source, description).
5) מצות קרבן התמיד בין הערבים - The daily afternoon Tamid offering (source, description).
6) מצות עבודת הכהנים בקרבנות - The Temple Service of the Kohanim with the offerings (source, description).
7) מצות אכילת פירות שביעית - To only eat Shemittah fruit (source, description).
8) מצות ישיבת הארץ - To settle the land of Israel (source, description).
9) מצות לקיים שבועה של רשות - To fulfill a vow to do something voluntary (source, description).
10) מצות להניח אחת מן הרוחות בלי מצור בשעת מלחמה - To leave one side of a city during a siege unprotected (source,description).
11) מצות זכירה בפה מעשה מרים - To recall with words the story of Miriam (source, description)2.
12) מצות להיות תמים עם השם - To be with pure trust trust in Hashem (source, description).
13) מצות לפרוש תרומה מן היפה - To separate Terumah from the best portion (source, description).
14) מצות להקריב רק בקר כשבים או עזים - To bring as an offering only cattle, sheep, or goats (source, description).
15) מצות להקריב כל קרבנות בין שני התמידין - To offer all offerings between the two Tamid offerings (source, description).
16) מצות אכילת הפסח בלילה - To eat the Pesach offering at night (source, description).
17) מצות גואל הדם להרוג הרוצח במזיד - For the Goel HaDam to kill an intentional killer (source, description).
18) מצות להרחיק מלהנות בבגד או בית שנראה בו נגע - To avoid benefiting from a garment or house which has an affliction (source, description).
19) מצות בירכת התורה - To make a blessing on learning Torah (source, description).
20) מצות להחיות גר תושב - To support and sustain a Ger Toshav (source, description).
21) מצות להחזיר רבית - To return interest (source, description).
22) מצות להרחיק מלהנות משער נזיר - To avoid benefiting from the hair of a Nazir (source, description).
23) מצות נתינת תרומה לכהן - To give terumah to the Kohen (source, description).
24) מצות נתינת מעשר ראשון ללוי - To give Maaser Sheni to the Levi (source, description).
25) מצות נתינת מעשר עני - To give Maaser Ani (source, description);
26) מצות נתינת חלה לכהן - To give Chalah to the Kohen (source, description).
27) מצות לעבר השנה - To calculate and establish leap years (source, description).
28) מצות עשיית הארון והכפרת - To make the Aron and its cover (source, description).
29) מצות ב"ד להרוג מי שמחוייב מיתה - For Beis Din to kill those who are liable to death (source, description).
30) מצות לחכם להורות בלי להיות שכורים - For a Sage to rule without being intoxicated (source, description)3.
31) לשחרר עבד כנעני אם הפיל עינו או שינו - For a master to free his Caananite slave if he removed his eye or tooth (source, description).
32) לתת לבכור פי שנים - To give a first born a double inheritance (source, description).
The Ramban disagrees with the following negative mitzvos:
1) [#5] שלא להשתחוות לעבודה זרה - Not to bow to idols. Because this is included in Rambam's #1, not to believe in idols.
2) [#6] שלא לעבוד עבודה זרה במה שדרכה להעבד - Not to worship idols in their normal mode of service. Because this is included in Rambam's #1, not to believe in idols.
3) [#28] שלא לשמע ממתנבא בשם עבודה זרה - Not to listen to someone who prophecizes in the name of an idol. Because this is just another verse prohibiting idol worship.
a) [#53] שלא ישא עמוני ומואבי בת ישראל - An Ammonite or Moabite cannot marry a Jewish woman. The Ramban splits this mitzvah into two, this one being about an Ammonite. The other is below, addition #4.
4) [#58] שלא לערוץ מפני אחד מכל שבעה עממין - Do not be afraid of the enemy during battle. Because this is a promise not a command.
5) [#69] שלא יכנס בעל מום בהיכל - A blemished Kohen shall not enter the Temple Heichal. Because "he shouldn't enter" refers to doing the Temple service, which is already in mitzvah #70.
6) [#71] שלא יעבוד כהן בעל מום עובר - A Kohen with a temporary blemish should not perform the Temple service. Because this is just part of mitzvah #70.
7) [#95] שלא להקריב קרבן בעל מום עובר - Not to sacrifice an animal with a temporary blemish. Because this is just part of mitzvah #91.
b) [#98] שלא להקריב שאור או דבש - Not to offer leaven or honey. The Ramban splits this mitzvah into two, this one being about offering leaven. The other is below, addition #8.
c) [#100] שלא להקריב אתנן זונה ומחיר כלב - Not to offer a harlot's wage or the exchange for a dog. The Ramban splits this into two, this one being about a harlot's wage. The other is below, addition #9.
d) [#125] שלא לאכול הפסח נא ומבושל - Not to eat the Pesach offering raw or boiled. The Ramban splits this into two, this one being about a eating the Pesach offering raw. The other is below, addition #10.
8) [#142] שלא לאכול מעשר שני של דגן חוץ לירושלם - Not to eat Maaser Sheni of grain outside Jerusalem. The Rambam counts these as three mitzvos (#142-144), but the Ramban combines them all into one.
9) [#143] שלא לאכול מעשר שני של תירוש חוץ לירושלם - Not to eat Maaser Sheni of wine outside Jerusalem. The Rambam counts these as three mitzvos (#142-144), but the Ramban combines them all into one,
10) [#150] שלא לאכול מעשר שני בטומאה - Not to eat Maaser Sheni while impure. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
11) [#151] שלא לאכול מעשר שנע באנינות - Not to eat Maaser Sheni as an Onen. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
12) [#152] שלא להוציא דמי מעשר שני בשאר דברים שאינם מאכילה ושתייה - Not to purchase with Maaser Sheni money anything besides food or drink. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
13) [#177] שלא לאכול מן השרצים המתהוים מן העפוש - Not to eat bugs that come from sweat. As per Ramban's understanding of Shoresh #9, if it's essentially one prohibition, it doesn't matter if there are many multiples of lashes.
14) [#178] שלא לאכול מיני שרצים דקים הנולדים בזרעים ובפירות - Not to eat worms that come from seeds or fruits. Because of #13.
15) [#179] שלא לאכול משרץ המים - Not to eat sea bugs. Because of #13.
16) [#190] שלא לאכול קלי מתבואה עד ביום ההוא - Not to eat roasted grain until the Omer is brought. Because of #13.
17) [#191] שלא לאכול כרמל מתבואה עד הזמן הנזכר - Not to eat fresh grain until the Omer is brought. Because of #13.
18) [#194] שלא לשתות יין נסך - Not to drink an idol's wine libations. This is the same as mitzvah #25.
19) [#198] שלא לאכול מכל דבר שיש בו חמץ - Not to eat anything that has chametz in it. This is just an extra prohibition against chametz, so it's not worth counting again.
20) [#199] שלא לאכול חמץ אחר חצות - Not to eat chametz after noon. This is the opinion of R' Yehudah, but we hold like R' Shimon that there's no negative commandment on this.
e) [#256] שלא לענות יתום ואלמנה - Not to oppress an orphan or widow. The Ramban splits this mitzvah into two, this one being about an orphan. The other is below, addition #21.
f) [#290] שלא לחתוך הדין באומד הדעת - Not to decide someone's fate just with estimation. The Ramban splits this mitzvah into two, this one being about killing someone without proper testimony (see Megillas Esther ad. loc.). The other is below, addition #22.
21) [#294] שלא לענוש האנוס - Not to punish someone coerced. The Ramban holds that this is just another case of shelilus, that they aren't punished, not that it's a command.
22) [#307] שלא לגלח שער הנתק - Not to shave the hair of a nesek affliction. This is the same as #308, and no need to count it as two mitzvos.
23) [#318] שלא לקלל אב ואם - Not to curse one's father or mother. This is the same prohibition against cursing any Jew, despite having a different punishment.
24) [#319] שלא להכות אב ואם - Not to injure one's father or mother. This is the same prohibition against hurting any Jew, despite having a different punishment.
The Ramban replaces them with the following negative mitzvos4:
1) שלא לשכוח השם - Not to forget Hashem (source, description).
2) שלא לשכוח מעמד הר סיני - Not to forget the giving of the Torah at Sinai (source, description).
3) שלא לשנות הסדר של כלי המקדש - Not to change the order of the Temple vessels (source, description).
4) שלא ישא מואבי בת ישראל - A Moabite cannot marry a Jewish woman (source, description).
5) שלא לשחוט קרבנות על מנת לאכלם או לעשות עבודתם חוץ למקומם או לזמנם - Not to slaughter offerings with intention to eat them or perform their service outside of their location or time (source, description).
6) שלא לאכול קרבנות ששחטן על מנת לאכלן או להקריבן חוץ למקומן - Not to eat offerings which were slaughtered with intention to eat them or perform their service outside of their location (source, description).
7) שלא לאכול הצפור ששחטו אותה לטהרת המצורע - Not to eat of the bird slaughtered for the metzora's purification (source, description).
8) שלא להקריב דבש - Not to offer honey (source, description).
9) שלא להקריב מחיר כלב - Not to offer the exchange for a dog (source, description).
10) שלא לאכול הפסח מבושל - Not to eat the Pesach offering boiled (source, description).
11) שלא ירימו הלויים תרומת מעשר מן הרע שבו אלא היפה - For the Leviim not to separate Terumas Maaser from the bad, rather from the good (source, description).
12) שלא נקבל שקלים מן הגוים ולא נצרף אותם עמנו בקרבנות הציבור - Not to accept money from non-Jews or combine them with ours for communal offerings (source, description).
13) שלא לקבל עדות מיוחדת בדיני נפשות - Not to accept non-unified testimony for capital cases (source, description).
14) שלא יבוא האיש הירא ורך הלבב במלחמה - For a fearful person not to join in a war (source, description).
15) שלא לעשות דבר שיסלק השכינה בזמן מלחמה - Not to do things which remove the Divine Presence during wartime (source, description).
16) שלא להעביר הבכורה מן הבן הגדול - Not to withold the double inheritance from a firstborn (source, description).
17) שלא לחמול על נפש הרוצח במזיד - Not to have mercy on an intentional killer (source), description).
18) שלא יבא החולץ על יבמתו - For someone who performed chalitzah not to lie with her afterwards (source, description).
19) שלא יבוא איש על אשתו שזנתה תחתיו - Not to lie with one's wife who committed adultery (source, description).
20) שלא יחר בעינינו בשלחנו העבדים חפשים בשנה השביעית - Not to be upset about seeing our slaves free in their seventh year (source, description).
21) שלא לענות אלמנה - Not to oppress a widow (source, description).
22) שלא להרוג צדיק בדין - Not to execute someone who was acquitted (source, description (see also Megillas Esther ad. loc.).
23) שלא יחר בעינינו בתתנו צדקה לעניים - Not to be upset about giving tzedaka to the poor (source, description).
24) שלא לגנוב כלי שרת מכלי המקדש - Not to steal any Temple vessel (source, description).
25) שלא ישא לוי ארון הקדש בכתף מבן חמשים שנה - For a Levi not to carry the Aron Kodesh on his shoulders from the age of fifty (source, description).
26) שלא תחטיא ב"ד הארץ להניח מחזיר גרושתו - For Beis Din not to ignore someone who remarried his divorcee (source, description).
27) שלא יזנה איש הארץ בזנות - For a man not to profane the land with illicit relations (source, description).
28) שלא יחלל כהן גדול המקדש בצאתו מהעבודה מפני אבל - For a Kohen Gadol not to defile the Temple by leaving his service due to mourning (source, description).
29) שלא תטמא ב"ד את הארץ בלינת המת - For Beis Din not to profane the land by leaving a corpse hanging (source, description).
30) שלא לחלוק על הכהונה - Not to dispute the Kehuna (source, description).
31) שלא להנות מעגלה ערופה ופטר חמור - Not to benefit from the beheaded calf or firstborn of a donkey (source, description).
1 There seems to be a typo in Derech Mitzvosecha, which says the Ramban split mitzvah #247 into two.
2 The following mitzvah in Shichchas HaAsin LeDaas HaRambam #6, has the mitzvah of מצות לאכול מן האילנות שבגבולה כל ימי המצור ולא לכרות אותם - To enjoy the fruits of the trees surrounding a city under siege and not to destroy them (source). The Mishneh LaMelech claims that the Ramban listed it only according to the opinion of the Rambam, who himself should have listed it. However, the Ramban himself doesn't hold it should be included, for the same reason as #24 above. This explains why the Ramban at the conclusion of his Hasagos says there are only 16 mitvos that the Rambam forgot, when he himself listed 17.
3 At the end of Ramban's Hasagos, he says that he's unsure if he should list the positive mitzvah for a King or Shofet to ask the Urim VeTumim if the people should go out to war. He says if it shouldn't be counted, then he'll count the mitzvah for a Sage to rule while not intoxicated. He leaves it as an unresolved question, but in his Hasagos to negative mitzvah #73 he pretty confidently counts the latter as a mitzvah, so I wrote above only that one. However, at the end of the Zohar HaRakiah he writes that we cannot rely on what the Ramban writes in his Hasagos; only on what he writes at the end when he sums up the mitzvos. This is also brought in the sefer Seder Hamitzvos.
4 The Ramban in his summary at the end of his Hasagos writes that the Rambam is missing seven negative mitzvos. Apparently he had a different version of Sefer HaMitzvos than we have. The Zohar HaRakiya tries to figure out which were missing, and feels he discovered five. However, Rav Chaim Heller (Yevul HaYovlos pg. 82) disagrees, and feels there was simply a copyist error at fault. He therefore says it's a waste of time trying to figure out which mitzvos were missing from his version. Regardless, the Ramban therefore added 7 more to the 24 he removed.
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