Computer GPU

definitely the 1060.

1060 all the way, no real choice there.

Today I bought 1060 6gb and it is absolutely amazing(Also got AC: oriins), so I'll recommend it

Cant help between these 2, but I know that when Ill be buying a new GPU, I wont be buying anything under GTX 1070. (Ive got a GTX 750Ti currently).

What are the prices for those 2 in your country?

never buy old video cards if you have the option to pick a new one, the 580 is from 2010, that's like a million years in video cards evolution.

1060 is way better. One benchmarking utility has the GTX 580 at 5,022 and the 1060 at 8,808. That's 75% more graphics power on that benchmark...

Edit: But I think you are talking about the Radeon RX 580 and not GTX 580 which benches at 8,253 which is at lot more comparable...
I'd still go with the 1060 though, but it's a closer decision.

Games that utilize more than 6 GB of graphics memory might be better on the 580, but I think most games will be better on the 1060.
And then there is whether a game is optimized for AMD or NVIDIA...

I don´t think there is a Sapphire GTX 580 8gb. Perhaps you mean Sapphire Nitro+ RX 580 8gb from AMD. Both the 1060 and the rx 580 are good cards, they perform about the same. Power wise and heat wise I think the 1060 is better, specially the gaming x from msi.

the 580 hasn't even half the graphic processing power of the 1060..
don't buy the 580 unless you get in for about 10€ including shipment..

As others have stated you probably mean the RX 580, the GTX 580 is sub par for gaming.

Choice is yours though. I am going to go with the GTX 1060 myself I think, as the extra cost in electricity for relatively the same performance of the RX 580 doesn't seem worth it.

Here is a Techspot article comparing the two on a bunch of games...
https://www.techspot.com/review/1393-radeon-rx-580-vs-geforce-gtx-1060/
It looks like a toss-up. Some games are slightly better on one than the other. Maybe see if they list games you are planning to play and take the one that performs best on those...

Checking some reviews (example), the RX 580 performs around the neighbourhood of the 1060. It beats it in some cases, loses in others. Different reviews show somewhat different results, which I guess have to do with the particular cards tested and the driver versions, but I'd say they two cards are in the same ballpark.

The RX 580 will use more power (although it likely can be undervolted to reduce power use, which is easy to try with the AMD drivers).

The only major advantage I see for the AMD card is FreeSync, which is becoming quite common in monitors these days.

The cheapest one unless you have a freesinc monitor

Option A: Go for the cheapest, since they are nearly the same. Most importantly, 8GB of VRAM is a total overkill for the RX 580, so conside buying the 4GB version (unless you really want to have a future-proof amount of VRAM on a not-so-future-proof card).
Option B: Get the RX 580 to support AMD, especially if you have a FreeSync monitor or if you will get one in the future; again, both cards are very similar.
Option C: Get the one of the new models for the 1060 6GB version, Nvidia got nice additional software like ShadowPlay and Ansel, and the 1060 is A LOT less power hungry.

If it wasn't so power hungry, personally I would go for the RX 580. I got the 1060 at launch, which was slightly better of the 480 (and around the same prince in my Country) on most aspects, but right know the 580 is overall slightly better. Less power = less heat = [usually] less noise, less need to invest in efficient cooling solutions, and of course a lesser impact on the electricity bill if you play many hours of intense games daily.

Personally I'd go for the RX580.
Freesync, so cheaper monitor if that's interesting for you.
No Geforce Experience registration crap, dick move by Nvidia
The power consumption can be an issue, depending on how much you actually play.

Got a RX480 Saphire 8GB vram, I'm happy with what I have and I recommend it
Since the RX580 is an upgrade from the RX480, I'd go for that

The RX 580, unless you plan to get a new card in 2-3 years or so. In that case, the GTX 1060 is a better purchase. Of course, a Vega 56 would be the best, if you could find one (and would not cost twice as much)…

GTX1070 at LEAST
If you can't afford it then wait for the next gen to come out and prices will drop on current stuff.
Or get what you can afford of the new gen when it comes out..

IMHO the GTX1060 may struggle in a few months when newer games come out, where as the GTX1070 will be a more future proofed card for longer..

For gaming which one would be better in the long run for gaming, power wise and heat wise?

1060

They are more or less the same. I'd personally go for the 1060, simply because more games are optimized for the Nvidia cards than for AMD's. The extra 2GB of VRAM is good, however it's irrelevant when playing on 1080p for which this card is meant. You'll run out of computing power way earlier than you'll run out of VRAM.

Check some sites that use actual benchmarks and FPS etc to see who comes out on top?

I've bought XFX RX 480 8gb GTR over the gtx 1060 gb one year ago. So far it's running great.
If I were you, I will definitely buy RX 580 8 Gig for sure. :D

if rx 580 has 8gb vram I'd go for it, nvidia is starting to rip-off people

If you can get the 580 cheaper and already have a PSU good enough for it, id buy it... performance wise they will be pretty damn close (some games might run better on 580, some might run better on 1060 but overall by a small margin) and if your going to be gaming at 1080p they should give more than enough fps in max settings. Temp wise i'd say the 580 should get you the lower temps however it could be just by a few degrees.
As a bonus with AMD you can get a freesync monitor way cheaper than a g-sync, however g-sync from what i've heard is the better one.

Idk im intel hd user atm.. :(