What games do you plan on playing? Both cards are pretty good as you said, the gtx 560 has an edge, however what settings do you plan on gaming on? What resolution? And anything specific you plan on using it for or just gaming, and everyday use?
GTX 560 and the HD 6870 are close in terms of performance. At times the 560 wins and at other times the 6870 wins.
I'll take the HD 6870 for $130 because that's really cheap, considering it costs around $160.
Also, it consumes a lot less power than the 560.
$20 more for a 560 is tempting, but if you have the money for it, it can be considered as well.
I mainly play RPGs (Skyrim, Mass Effect, The Witcher, plan on picking up Diablo III, etc) and every now and then some Civ V. My biggest concern with this particular 6870 is that I've read a lot of reviews saying it runs hot and the fan is loud, which are dissuading me from buying it. I'm thinking that even though the GTX 560 is pretty even or only slightly better than the 6870 that the $20 is worth not having to worry about my card overheating.