The ROG Strix is the best 1060 you can buy. But the fps difference between the two would be minimal.
Between the two you are only looking at three things; Overclocking out of the box, Memory speed out of the box, and overclock potential. In both of those Asus wins with a higher clock speed and what appears to be better cooling, but without looking at the individual PCB's I couldn't say for sure. But at this point it won't make too much of a difference. You'd see maybe 1-2 possibly 3 more fps over the Gigabyte. Price to performance I would choose the Gigabyte, but I really don't like Gigabyte products so if I were limited to that price, I would go for a Zotac AMP! or an MSI Gaming X at that price, but its pretty much just aesthetics, neither of those will really perform that much worse than the Gigabyte.
But what is important to me most is speed out of the box, which is why I got a Zotac AMP Extreme 1080. I really don't want to mess with the stable settings on my card, but I regularly get 2063MHz speeds stable out of the box, whereas many overclock to 2150MHz with less quality cards after tweaking the settings and that appears to be the max. Most cards have a similar overclock potential but you have to mess with the settings to do it. If you are good at that stuff you could literally get whatever card you want, but if you aren't and you want the best out of the card box, its the Asus ROG Strix for the 1060.