Same performance
The RX580 is cheaper (as low as $155 at newegg.com in the US)
The RX580 draws more power while running full-tilt**
AMD = FreeSync and Nvidia = GSync. FreeSync is cheaper than GSync and you have options for monitors with lesser specs being cheaper still. FreeSync & GSync are both functionally the same.
**Where the GTX1060 will always run at 100+% clockspeeds while gaming, AMD drivers have Frame Rate Target Control and Chill which save a lot of power in lighter games.