XFX cards are generally more expensive then the other brands, however my experience with HIS has not been positive and if you were going down either the AMD or Nvidia route, go with an XFX Double D or a MSI Twin Frozr card. My first GPU that I ever bought was made by them, a 5770 and the fan made a lot of squeaking and whirring noises, it was pretty much an overpriced reference model to be perfectly honest.
The 6770 is reasonably smaller then the 6870, though the 6870 isn't a huge card, however you should be considering the fact that the EVGA 560 happens to be Superclocked, so it comes stock at higher speeds, however regardless it's still more powerful then the 6870. I have a MSI Twin Frozr 6870 and really like it, in the right case it's not super loud with the fans cranked up.
Go for something with dual fans, it's going to last longer and cool better, don't skimp out on single fan GPU's these days.