BF3 is for sure in favor of Nvidia. So is Skyrim. For me, it seems that the more popular games out at the moment favor Nvidia. The ones that favor AMD seem to be less popular ones. Does anyone actually play Aliens vs. Predator?
I would go with the 560 in order to get PhysX, Adaptive VSync, FXAA, forced Ambient Occlusion, and better driver support for new games.
If you were overclocking (and I see that you aren't), this question would actually be pretty easy, the 6870 has very little headroom to overclock, the 560 has a lot of overclocking headroom.
But since you aren't overclocking, and I see that EVGA you listed is out of stock, why not go for the Gigabyte GTX 560 Superclock? $140 after rebate and a core clock of 900 mhz means your overclock is built in at the factory.http://www.newegg.com/Product/Product.aspx?Item=N82E16814125390