Based on reviews I've seen, an RX 580 is perhaps slightly faster on average, though it depends on the game, and in practice both cards should offer a very similar level of performance. A 1060 will be somewhat more power efficient though, and in turn put out less heat and cost a bit less to operate, and might be a better fit for your existing power supply. Personally, at least as far as gaming goes, I wouldn't spend more on a 580 than a 1060 6GB, unless perhaps I was pairing it with a FreeSync Display.
As for mining, I'm not really into that, but it seems payouts have been on a downward trend for months. I'm not sure that there will be a whole lot of money to be made there, and it's likely that it could put added wear on components.
And that resolution is relatively high, working out to more than 1440p in terms of total pixels. For many newer games at high settings, you might want to look at a faster card, like a 1070. Or perhaps wait until sometime later this year, when there will likely be a successor to the 1060 coming out, that I suspect should offer 1070-level performance at a lower price point. The current graphics card designs from AMD and Nvidia are around two years old at this point, and they would likely be priced a fair amount lower right now if it weren't for strong mining demand earlier in the year resulting in shortages. Technically, the 580 came out a year ago, but it's pretty much just a higher-clocked minor revision of the 480. It was possible to get RX 480s for well under US $200 a little over a year ago.