If you don't mind overclocking and the extra power consumption that will come out of it vs. the 6950, I'd lean towards the GTX560.
These are my reasons:
1) At your resolution, 2GB of Ram won't be beneficial
2) 6950s don't have a lot of overclocking room
3) 6950 2GB that unlock are MIA, or command a price premium above $230
4)
GTX560 @ 950mhz is as fast as a GTX570 (and since #3 is not an option, you can't really get a 6950 1GB to be as fast as a 570).
5) NV does better in Civ5, Crysis 2, Starcraft 2 (some key games you may play). Obviously if you play F1 2010 or Dragon Age II, I would lean towards AMD. Moreso, NV performs better at lower resolutions (but your CPU would need to be overclocked to get the most of that since NV cards are more CPU dependent).
6) Some excellent 560s are available with aftermarket coolers, or you can go with EVGA for their premium lifetime warranty and 90-day step-up.
All in all, there are some excellent bundles to be had if you look hard enough. For instance, MSI Hawk @ 950mhz with a Hawx2, Just Cause 2, Mafia 2, nba2k11, 3dmark11 for
$233.
That's GTX570 performance with a bunch of software for $30 more than a 6950 1GB. If you don't care for any of that, just pick whichever card you can find cheaper for $200.