I'd actually been looking specifically at the 760 and 770 as potential choices, wasn't quite sure myself why I'd wanna pay similar money for the 580. Also I was looking at the specs for games I plan on playing primarily with it, and at least one says for optimal performance on an Nvidia card it should be 600 series or higher (not sure if this is just a marketing technique or something but it came from the devs so...) I've also heard several horror stories about AMD drivers and compatibility issues with them and games before, which is not to say Nvidia doesn't have its own problems, but I've just heard more problems stemming from AMD cards personally.
Had AMD for 4 or 5 years myself, only ever had a single problem with their drivers and it requires 4 monitors to reproduce. Most/all driver issues with AMD I see is from people using extreme configurations, usually several cards installed.
I also think the driver version naming is also very straight forward compared to my experience with Nvidias system. AMD simply names them after year and month it was release - The current one is 13.9 for September 2013 for example.
If a game says it runs better with one or the other it's mostly marketing. The difference in performance is fairly minimal in almost every single game. Usually the developers are paid/sponsored by either Nvidia or AMD to promote them by optimizing the game for their cards. I've even seen games made to run best with Nvidia cards where AMD cards run notably better and vice versa.
As said once more, AMD's HD7970 and R9-280X is currently priced at about $100 less than the 770 but is more or less on par in terms of raw performance.
Nvidia has a lot of features going for them though, they have CUDA (which only matters really if you edit videos and such using Adobe software), Physx (adds some fancy but mostly useless effects in supported games), G-sync (makes things look better and more natural, but requires you to buy a supported monitor) and a few more things.
AMD has things like Mantle (gives increased performance in supported games, so far only BF4 is on the list), OpenGL support (an open-source alternative to CUDA) and generally better potential for overclocking.
Both of them offers free game bundles;
Nvidia gives you Assassin's Creed IV and Splinter Cell: Blacklist with the 760 and adds Batman: Arkham Origins to the list with the 770.
AMD, with the 7970, lets you choose any 3 games from this list: Saints Row IV, Tomb Raider, Sleeping dogs, Deux Ex: Human Revolution, Devil May Cry, Far Cry 3, Far Cry 3: Blood Dragon, Dirt 3, Dirt Showdown and Sniper Elite. (Although the R9-280X is currently excluded from this offer)
I'd say honestly that it comes down to if you want the more premium features Nvidia offers for a higher price or a better value and equal performance at a lower price which is what AMD stands for.