Every time I think of computer nerd, I get the mental image of some one that have thick glasses and have a fashion sense that his/her mother still dresses him/her in the morning.
Computer geeks tend to have a little or a lot more fashion sense than a computer nerd.
Computer nerds tend to take pride in the fact that they still can find a use for that old Pentium 100 that you are about to throw out. To them, this most likely going to be an upgrade to what they currently have. However in Nerd defence, the fact that they have kepted that old 486 running as long as they did is an achievement in itself.
Where as a computer geek will most likely be making ure that they have the latest and greatest computer.
Nerd are more likely to be using linux and various other Free-BSD software. Where as geeks tend are more likely to have that pirated copy of Windows Ultimate.