So aside from a very opinionated answer, Here are the actual facts, even though this is an old post this is literally the first thing that pops up on Google when you type in "Asus ROG Strix vs. Zotac AMP Extreme." I have used Zotac before and I can assure anyone reading this they make fantastic cards. The 1070 versions of these cards are almost exactly the same, minus the software used to run them, both still have stop fan technology which means neither's fans will spin up under a certain temperature, this can easily be fixed with a custom profile, however I know for a fact in the case of Zotac the Firestorm software and any other software will clash trying to use the pre-programed stop-fan technology, and setting the fan speed anything less than 30% will give a very annoying whirring cycle. The best way around this is to create your own stop-fan profile and simply never have the fans set less than 30%. And with my system running 5 Corsair Magnetic Levitation 140mm fans at a very low 900rpm, and also have the dead silent Seasonic Prime PSU, my entire system is dead silent, and the fans are most certainly NOT audible at 30%, at 40% there is a tiny whir but still barely audible, the fan does not truly become audible until 50%, and again as a user (of the 1080 but minus the chip and overclock it is physically identical to the 1070) you will hardly ever need the fan past 50%, my card almost never goes over 55C and it is NEVER over 60C, the cooling is absolutely fantastic if you can deal with the noise. I cannot vouch for Asus in this respect, but I know the specs between these two cards are identical.
As for the power consumption... Yes the Zotac is specced for more, but that is because it is designed to run HIGHER than the advertised clock speeds. Again as an actual user I regularly see my 1080 AMP Extreme well over 2000MHz clock speed when it is technically only specced for 1911MHz, That's part of what make it the fastest 1080 on the market, and actual fps benchmarks show this. Out of the box performance clearly bests ANY other 1080 available, the only thing needed to do is get rid of the bad fan profile which let this card run pretty hot, and set it up with a good profile, and it runs extremely fast, very cool, and somewhat silent, and to be honest, if you game with a headset you can actually run the fans up to 70% and not hear them.
But no despite the companies customer support, I would not know I have an Asus motherboard, and my Zotac card, both make absolutely fantastic products and the chance you will actually need to use their CS is low.
Bottom line, both cards are pretty much IDENTICAL and aside from FPS benchmarks which will show a max of 1-2 fps difference, most likely favoring the Zotac, and the fan properties can easily be fixed with MSI:Afterburner or another similar software with a custom fan profile. It's mostly a choice between which looks better if you don't mind the dismal performance difference IF ANY, and the slightly less cooling capacity of the Asus due to the clearly smaller heatsink, which again, would only be a very minor difference.
Anyways this is for anyone actually looking into these two cards and want to gather all the facts and not just take the word of the first person to trash one of the companies, because he either had a bad experience or doesn't know what he's talking about.
Asus and Zotac are both top notch and make fantastic products and have at least good CS, again I wouldn't even know I have never had to use either since I have never had a problem with their products.