what kind of motherboard do you have? if its an older pci card, then youll need to change the motherboard too. but by the looks of your processor, it seems like a rather new computer, so i would assume it has an agp slot.
You dont need to buy a new case for a more powerfull (its not powerfuller :P ), you should be able to just replace it, and hook everything up.
The first graphics card you linked to;
is better. It's a 6800, which is a bigger number then 6600, so therefore its bigger and better (most of the time). It's a pretty damm good card, so you should be satisfied with it.