grammar - Differences between ways to express obligation?

1) I know that, to express something that you must do, you can use 方{ほう}がいい and, for a stronger feeling, べき.

2) I am also aware you can use conditionals like 〜なくちゃ、〜なきゃ、〜ないと to express that you need to do something.

Though I understand the difference between 方がいい and べき, and I somewhat understand the differences in 〜なくちゃ、〜なきゃ、and 〜ないと, I don't understand the difference between those two groups. I don't understand the difference between 1 and 2.

Is there some sort of nuance I'm missing? Also, is there a difference between something you need (by yourself) versus what you're obligated to do (by society or rules or something)?