Most of the time they can be used interchangeably to express something you "want" (vouloir) or "have a desire for" (avoir envie de). There's a slight difference in how concrete or direct your wish is, which is also reflected in the corresponding English phrases. Vouloir is more direct than avoir envie de.

But there are occasions, where you'd only use one or the other. For example:

J'ai envie de vomir. -- I am going to throw up.
J'ai envie d'aller aux toilettes. -- I need to go to the bathroom.