I am not sure which one I should pick for my PC, is there a LOT of difference between the two? And I am more attracted to the AMD because I will get a lot of extra's with it (3 games: crysis 3, farcry 3, bioshock inf.) and it costs less. I am a casual gamer. I enjoy playing games but I don't demand extraordinary performance.
Here's a review of the Two GPU's focusing on them at 4K res but if you ignore the writing about 4K/high res and look at the charts you'll see the two benchmarks for games at 1080p including one chart for Farcry 3 the game you listed. The 280x costs less than a 770 but performs similarly. I personally would wait until aftermarket companies have gotten their hands on the 280x and changed the stock fan, in the UK this is apparently in 2-5weeks but I don't know about your country.