Most Helpful Guy
To me, a geek is someone who is very interested in a particular topic and spends an inordinate amount of time, energy and money on that topic. It could be politics, music, art, reading, science or whatever.
A nerd, on the other hand, is someone who has an excessive amount of knowledge and/or skill in a particular area.
Most Helpful Girl
Maybe a bit romanticized, but:
To me a geek is someone who's playful and passionate. A geek is someone who loses himself in the story of a book, or gets absorbed in a game. Someone who creates worlds and epics playing D&D or dwells in middle age. Someone who dreams about the future and everything that's possible and impossible.