As far as I know both means the same thing. Except, なぜ is much more polite than どうして。 Even though many will say that their are absolutely no difference at all, I myself tried to figure out a perfect translation for both of them with the help of watching/listening to a lot of Japanese conversations from movies, dramas etc. I think なぜ is the exact same word for "Why" in English, whereas どうして　would mean "what for" / "for what" I hope you could understand the explanations. If not please ask me again and I will try my best to help you...^_^

ありがとうございます。＾＿＾

originat; どうして from どう(how)＋して(do).　　　なぜ漢字is何故。何＝what、故＝reason but now 　　どうして=なぜ＝why (^.^)