Mixed feelings..
1) In theory, the RX-580 should do a lot better, relatively, as time goes on and its architecture is optimized for, but
2) NVidia tends to have better driver support. In fact, I've heard relatively more claims that the RX series is having issues than on the NVidia side.
Now more than 4GB not being needed is mostly true, though it's impossible to predict that over the next few years.
So if the price is similar I'd go with the GTX1060 6GB.
Here's a modern comparison too:
https://www.techpowerup.com/reviews/Sapphire/RX_580_Nitro_Plus/30.html
They are actually closer to EQUAL if we compared cards with the same OC and cooler. Again, the RX-580 will do a little better in some modern games, and the GTX1060 fairs better on average in older games.