First off, let me start by saying this: I want as much horsepower from the card without overclocking as possible. That means, whichever card will give me the most FPS at 1080p in most games, that's what I want.
That really depends. You should read up benchmarks for whichever games you have or are planning to get, or the features you really want. Ex. Crysis 2 is optimized more for Nvidia, though today I'm not completely sure, BF3 for certain is for AMD.
Nvidia graphics cards will not grant you 3-way monitor setup on one card, though rarely anyone actually does this setup. AMD from HD5000 and up will, but does not support PhysX. If you want PhysX which is supported in few games, ex. Batman, Nvidia would be your choice.
generally between the two, the HD 6870 has a slight edge(its pretty small) but the 560 hits more people as one of the more talked about games is Battlefield 3, which the 560 has an edge over due to a preference in nvidia. other than that, on a non gaming use, AMD does better on opencl operations(bitcoin mining is a big example) while nvidia is built around programs that can use CUDA cores or physx.
I'm a programmer, I do a little Sony Vegas and Photoshop, planning on learning CryENGINE 3 SDK, and I play stuff like Fallout, Left 4 Dead, Call of Duty, Hitman, GTA, DiRT, Borderlands, and really, anything that comes out and looks like it'd be fun.
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.
Sony Vegas i think is an OpenCL program, so AMD will have an advantage on that side of the program. Photoshop will use CUDA (latest photoshop is trying to work with AMD and open CL) so Nvidia would have an advantage there. I think cryengine3 will use opencl, though i cant say that for sure.
*sigh* you guys really seem to be overthinking this. the op originally said he just wants the most fps @ 1080p. i compared these two cards back in november and they really are about the same (+-3%). so i say go with which ever one you can find the cheapest. fyi i only went with the 560 because i got it for around $150 w/batman ac for the free, if the 6870 was cheaper i would have got that. the end.
I have a feeling that the reason it turned into such an elaborate thread is because the two are so close in performance that the choice should be based upon whichever card has more features, compatibility, optimization, and lower price.