This has been the topic of much debate across the whole internet. If you really want to get into this we're going to have to get into the many definitions of each... and what sectors of the populace have which definitions... and the 10 minute slideshow presentation... and someone will (probably accurately) conclude I am both.
Here's the thumbnail version:
A geek is a person who is really, really into something. Like a car geek or a my little pony geek. My dad is a rock geek. Geeks can blend in with society fairly well. In fact you may not even know the person you're talking to is a geek unless you happen to hit his geek topic and he geeks out about it. Speaking of which, don't ever, ever, ever mention rocks to my dad. He will talk for hours and HOURS about each stone he has with him, how he cut it, how he polished it, what it is commonly used for...
A nerd is a computer geek.
Yup, it's that simple.