They are pretty close synonyms — there aren’t many nuances that differentiate them. But there are four notes to add:

1) “Vite” can also be an adjective, particularly (I am given to understand) in informal contexts. Edit: As discussed in the comments, your mileage among native speakers may vary when relying on this usage. However, I still see it taught in learning materials now and then.

The next three notes are derived from entries in le Dictionnaire de l’Académie française based on DVLF searches for “vite” and “rapide“.

2) Edit: Does not seem to be current. It suggests that “vite” is better suited to animals and other living things, as an adjective, anyway. That would make “rapide” the preferable choice for inanimate objects. I can’t say I’ve noticed a strong trend towards that distinction, and (edit) judging by Frank’s commentary below and the fact that the edition of this dictionary that was cited is 80 years old, it appears that this distinction has died out.

3) “Rapide” seems like the more suitable figurative choice for something is not literally moving fast, such as “un style rapide”, “un esprit rapide”.

4) “Rapide” may also carry connotations of brusqueness or impetuousness, similar to “hasty”.

Interested to see what the others think of these connotations.