word choice - Usage of たくさん vs. 多い

When do we use たくさん and when do we use 多い? I found both modifiers are used within similar sentences, for example:

人が多かったです
人がたくさんいました

I had this impression that 多い is used with countable nouns (i.e. similar to "many") while たくさん can be used with both countable and uncountable, but seems like it is used with "お金":

ふだんの生活で使うお金が多いからか――ということです

At first, I thought maybe in Japanese language "お金" is considered as countable, unlike the word "money" in English, but again I realized I was wrong when I found:

きょうは水が多いですね

Now I'm thinking that maybe the concepts of countable vs uncountable in Japanese are not as significant as in English. It seems to me that both たくさん and 多い may be used interchangeably.

Am I right? Or are there actually places where only one of them can be used but not the other?

EDIT:

Additional question: how are they related to potentially antonyms 少し and 少ない? Is 少し antonym of たくさん while 少ない is antonym of 多い, or vice versa, or it does not matter either way?

Answer

I don't think the existing answers are hitting this question from quite the right angle, so here is my take:

First, in sentences where you only wish to mention the presence of "a large number" or "a large quantity", 多い and たくさん can often be interchanged, provided you do some grammatical fiddling to get the rest of the sentence to work:

人が多いですね。 There are a lot of people here, huh?

人がたくさんいますね。 There are a lot of people here, huh?

多くの人が参加しました。 Many people participated.

たくさんの人が参加しました。 Many people participated.

But there's an alternate interpretation for the third sentence: 多くの could simply mean "a lot", but it could also mean "a large proportion". It's this idea of largeness in proportion or ratio that たくさん does not have. This shows up in sentences comparing one quantity to another:

今日は水が多いですね。 There's more water [than before] today.

これからは機会が多くなりそうです。 It looks like we'll have more opportunities [than before] from now on.

今年の春は去年の同じ時期より雨が多く降りました。 This spring, more rain fell than during the same time last year.

Note that 雨が多く降った is a legal sentence, but only within the proper context of comparing proportions, as demonstrated by the third sentence above.

By extension, 多い comes with a meaning similar to "more than necessary". This is another meaning たくさん does not have:

10人ならいいけど、20人はちょっと多いですね。 10 people would be good, but 20 is a bit much.

あのさ、これ、多くない？ Hey, isn't this too much?

You cannot replace 多い with たくさん in the above examples, because たくさん lacks the ability to compare a quantity to a certain level.

In response to your additional question about 少し and 少ない: Yes, you can think of these two as corresponding with たくさん and 多い, respectively. 少し only refers to quantity, but 少ない has the additional meanings of "small in proportion" and "less than necessary".