Nvidia and AMD both have their merits and demerits.
The 1060 3gb and 580 4gb are in the same price tier. 3gb vram won't be enough to max out eye candy in AAA games. So it's a plus for 580 4gb.
But most games uses Nvidia game works like hair and explosion graphics. And also some light effects.
Another icing on the cake for 580 is the availability of 144hz freesync monitors at the 200 bucks range. Consider the msi optix monitor. It's a great buy.