The HD7870 is very similar to a GTX660 though it VARIES slightly from game to game. The HD7870 however comes with a download coupon of Crysis 3.
Shadowplay on NVidia:
There are other differences including PHYSX but I think the only thing really worth mentioning is Shadowplay. This new software (coming in July?) can create streaming VIDEOS of the game you're playing with almost no performance hit (3% roughly). Previously methods using FRAPS or whatever could drop frame rates by 30% or more.
Shadowplay uses the NVENC video encoder in GTX600 and better GPU's so it's not something you can add in software to an AMD card unless it has similar hardware (which I didn't think it had).