Vakit is more like "moment" while zaman is basically "time". They are interchangable in most cases but there are situations they aren't; for instance, like Hande stated, when you talk about "time" concept in general, you cannot use vakit. (uzay-zaman, zaman makinesi) If you're not sure which one to use, prefer zaman.

They mean the same thing but zaman is more general while vakit is a little bit more specific. Zaman geçirmek=vakit geçirmek (spend time) Boş zamanım/vaktim var (I have free time) Zaman makinesi (time machine), but vakit makinesi is not correct.