No (*adjusts glasses, dabs at minor nosebleed*) they’re not the same.
Scientist and software engineer Burr Settles of Slackpropogation explains:
Both are dedicated to their subjects, and sometimes socially awkward. The distinction is that geeks are fans of their subjects, and nerds are practitioners of them. A computer geek might read Wired and tap the Silicon Valley rumor-mill for leads on the next hot-new-thing, while a computer nerd might read CLRS and keep an eye out for clever new ways of applying Dijkstra’s algorithm. Note that, while not synonyms, they are not necessarily distinct either: many geeks are also nerds (and vice versa).