Here's a really simplified answer...
Geeks usually delve into pop culture such as sci-fi, anime, fantasy, etc. Nerds are people who have a lot of book-smarts who get really into whatever field they're interested in. For example, my sister is an English nerd, but the more "famous" variants are into Mathematics and Science.
The two groups usually overlap... but you can get Geeks like myself who don't have much in the way of book-smarts and Nerds who don't like pop culture.