Like others have said these are a completely different class of GPU. The RX 480 will probably compete with the GTX 1060 in performance and price range just like the GTX 960 and R9 380 competed. The 1080 is two performance levels (and price levels) ahead of the the 1060 and 480.
The answer to your question comes down to your budget and resolution. If you are playing at 1080p the RX 480 and GTX 1060 will probably be great choices. The GTX 1070 should be able to max everything out at 1440p with 60 FPS.
I think the GTX 1080 makes sense if you want 4K or if you are shooting for 144 FPS at 1440p. It's also a good choice at 1080p/1440p for people who don't want to upgrade again for several years.