For everyday performance, there will be little if any difference between a 480/4, a 580/4 and a 480/8. They are basically the same card, all in the *80 performance range. You'd not notice or care that the 580 might get 5? Fps more, when you are already capable of fps up in the 100's+, beyond most standard monitors refresh. Detail settings will be identical.
The difference between the /4 and /8 comes into play only in certain games where they are optimized for higher vram. And it's a big difference. It's enough that it'll make or break a game at the detail settings you want. A massively detailed game like Witcher 3 or GTA:V, that's extremely fluid and fast moving is going to get bogged down at maximum settings on the 4G cards. You'll not only suffer fps loss, but to reverse the affects you'll need to drop detail settings. In the simpler, smaller games, that are more cpu dependent like WoW or flight Sims etc, it won't be an issue at all.
Having the extra vram isn't going to hurt at all, other than a few $ from the wallet. Not having the extra vram could possibly hurt in the long run if your gaming styles tend towards games requiring high vram.
As far as DX goes, 11 is different than 12, a lot more so than between 10 and 11. Most nvidia cards are designed around taking massive advantage of DX 11 and prior because that's what was the major avi of the games written. With DX12, those algorithms actually take better advantage of AMD's architecture, so it isn't so much that nvidia is not still doing its best, it's just that amd is finally catching a break and the card performance per slot is closing the gap. Back in the day, nvidia mostly had 3Gb cards for DX11, AMD was using 4Gb cards just to try and equalize the battle between like cards. Today, with DX12 titles, those older 4Gb amd cards are actually doing better now, the cards haven't changed, just the way they are used internally. With the 480/580 essentially being the same card, there won't be any real performance differences, just the little things like running slightly cooler or using less power, when judged against the exact same model