GTX 560 non-Ti = HD 6870 in performance. One leads the other depending on the games you play. Personally, I think the 560 is gaining the upper hand these days in newer games (Skyrim with latest drivers, Battlefield 3, Civilization V, RAGE, Dirt3, F1 2011, etc.). Still, the two cards are very, very close as you can see here:
AnandTech - Bench - GPU12
Counting only gaming benchmarks:
"Wins" for HD 6870 - 9
"Wins" for GTX 560 - 13 or 14 (depending on whether Civ V's compute test can actually be called a gaming benchmark)
I think the reasons for this have more to do with the success of the GTX 460 and 560 series combined, leading to many developers having a large common platform to test and optimize for (From Steam survey, GTX 460 + 560 are some of the most popular DX11 cards today). Whereas, AMD had a different architecture from 57xx to 58xx to 68xx.
I also think HD 6950 for it's price is better than any of it's competition for similar reasons. AMD also focuses it's driver testing on the 69xx series while NVIDIA focuses it's driver testing on GTX 460/560 series.