This is my summary of them:
RX480
1.)Appears to have better DX12/Vulkan performance
2.)More memory and memory bandwith
3.)History of AMD midrange cards faring better as time progresses(I had a GTX660 and GTX960 and can attest to that)
4.)FreeSync compatability
GTX1060
1.)Faster in DX11
2.)Better overclocking abilties although
YMMV
3.)Less power consumption(might not be a factor unless you play a lot each week)
4.)Cards appear to be more compact currently
5.)Better Linux drivers
6.)Want to use some specific Nvidia features(Shadowplay,etc and I am not sure how good the AMD equivalents are ATM)
On the balance of things,I probably would prefer the RX480 8GB for potential longevity but again another factor is what games you intend to play.