So with games like Assassins Creed and Tomb Raider coming out for Xbox One, does that mean that you can earn achievements on there and the 360 versions? Are there different achievements? This is all weird.
Well, XBOXONE has time-sensitive community achievements, but apart from that I think most games are just going to have the same achievements on each version of the game, which means that you won't be able to double up on them if you happen to play both versions. I could be completely wrong about that, though. Doing two sets of cheevos would be two times the work for the company, so I doubt anyone's going to go to any great length to do it.
If you want to play it twice then you can earn the rewards twice. They're two different products so they'd have to be separate sets of achievements. They'll be identical between versions 99 times out of 100 but getting an achievement in one version won't unlock the achievement in other version. The exception is cross buy games on PSN. Stuff like Spelunky, Divekick and Hotline Miami unlock across versions because it's a single purchase. Both versions read the same save file so they'd have to be joined.