The 480 and 1060 butt heads in most applications, but where the 1060 wins, it wins by a larger margin than when the 480 wins, so is the slightly stronger card overall. This applies to both sets, the lower ram and higher ram. But both will pretty much max out settings of all most every game at 1080p, so it boils down to preferences. Team Green or team Red. AMD has greater ability to crossfire as any board with dual slots is crossfire capable. You'll need a more expensive board to run sli, and the 1060 itself is non-sli capable, so crossfire is an option with the 480. Nvidia tends to have more stable drivers and less compatability issues. Nvidia has gpu accelerated physX, amd cards rely on the cpu for physX functions. Amd usually requires a larger psu vrs nvidia in the same gpu bracket. Etc...
Each has benefits and drawbacks, the card for you will be the one whose benefits apply to you the most.