EDIT: Here is another link with the full rig used listed at the top and a bunch of cards benchmarked.http://www.anandtech.com/show/4137/amds-gtx-560-ti-counteroffensive-radeon-hd-6950-1gb-xfxs-radeon-hd-6870-black-edition/5
That is benchmarks of both cards set side by side on an actual computer and it shows what quality settings are used
The site The Halo Don used states "Please note that the above 'benchmarks' are all just theoretical - the results were calculated based on the card's specifications, and real-world performance may (and probably will) vary at least a bit." underneath the pixel rate comparison.
Bdizzle11 is correct when saying it depends on the games you want to play. Certain games favor NVIDIA over ATI and vice versa. On the page I linked at the top you'll see what games prefer which brand and that can help you decide what card you want.
I personally have a XFX HD6950 2gb and I love it, but I have always preferred ATI cards