R9 280X is a much better choice than the 770.
Personally, I would get the R9 290. Actually, I wouldn't. I did get it, and it is fantastic. The reference model is loud though. If you need quiet, I would save some money at the same time and get a good aftermarket R9 280X or 7970.
If you're primarily a BF4 player, you have to keep in mind that AMD cards will get Mantle on BF4 in a month. Even without Mantle though, Nvidia cards are stuttering where AMD cards are smooth... Nvidia isn't really an option for brand agnostic, tech-savvy BF4 players right now.
Actually the 770 runs BF4 pretty solidly, I haven't noticed any performance problems (stability is a whole separate can of worms, but that's BF4 as it stands currently and buying an AMD card instead won't save you from that).
As per the recent [H] article following up on the stuttering and 7 vs 8.1, a lot of the nvidia stuttering seems related to win7. Win 8.1 is pretty much flawless (at least for playing BF4, I don't mean in general!). That's certainly a consideration if you're on win7 though - I haven't seen this personally but [H] have a pretty good track record on calling this sort of thing out. The game is also obviously unfinished (to the point I wouldn't really recommend it for a while), so a lot could change quite a bit on either side still (eg the recent nvidia drivers).
Right now though, both the 280X and 770 will comfortably max 1080p in 64p MP. Mantle is kind of irrelevant performance wise in this case even if it does turn out to be a huge increase as again, both these cards will already get great framerates at max detail anyway.
Long story short, if you're on win 8/8.1 buy whichever brand you prefer/which you can get the best pricing on/which game bundle you prefer, both are fine for max settings at 1080p in BF4. The 280X is a bit faster, but unless you're playing at higher resolutions or significantly increasing field of view, you won't be able to tell.
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