Naze is more formal than dôshite, and it only has the meaning "why." Dôshite, on the other hand, also has the meanings of "by what means/how," "rather/conversly (as in "he looks pretty weak, but he's rather/conversely strong"*), and also can be used to oppose or strongly deny some viewpoint. The latter usage I'm not terribly clear on yet.
*Ripped shamelessly from 大辞林 example sentence (3) under 一(副).