is neutral way of saying "Why/For what reason". This makes it different from なんで, なぜ
どうしても
implies that no matter what you do, something else holds;
it is persistent, and shows determination.
ロケットの推力がどうしても足りないんだって。 The thrust force of the rocket was insufficient, no matter what we did.
お前、どうしても日本に行く気か？ You intend to go to japan no matter what?
どうしてもこのアルバムを買わなければいけない。 No matter what happens, I must buy this CD!