Mini is size and has nothing to do with power requirements. There are mini gpus of 1060, 1070, 1080, etc, which still require the same stock power connections. They can also use all that power and better power delivery only matters when pushing the limits of ocing. Some have 2 fans like the 1080 mini yet are still relatively small. Smaller coolers does mean they won't cool as much, that's just how the world works, so they tend to not boost as high to keep temps down. But if you look at reviews, I've seen even the minis beating the stock reference design and those are a lot bigger.
If I had to choose, the minis tend to cost more and you'd be better off with a regular size.