With cards this close in price and performance, the most important question is what are your PC's specifications (including the monitor), and what games do you play or plan on playing?
In a vacuum, I'd go for the 1060 6GB over a 4GB 580. More VRAM, less power draw, and potentially quieter and cooler running, at least going by TDP and reviews. Drivers and other issues may favor AMD - I haven't kept up on what others have said about AMD vs Nvidia in that regard myself lately, though I haven't had issues with either company's drivers (at least not recently) - but I don't think that should be the major factor determining factor here.
If you look at it as $15 for an extra 2GB of VRAM (whether you need to use it now or not, it is extra headroom for the future) and greater efficiency, it's an easy decision. It would be a more difficult choice if we were talking about a less expensive 8GB 580, but alas, mining and memory pricing has pushed it up out of the discussion for this price-point.