This is a tough question since both cards perform admirably. The best reply comes from actual owners like myself, I've owned a 7870 and 2 7970.
I'm going to be returning my 2nd 7970 due to noise/coil whine. A major factor when buying these high performance cards is the aftermarket fans that come with it and the overall construction. The sapphire 7970 vapor x was LOUD at 60% fan speed and that wasn't even a crazy overclock, 1150/1600, it sounded like a blow dryer and it would have a very high pitched coil whine which you take a chance any card you purchase 7970 or 770 might have but I'm sure it's more prevalent in the 7000 series.
I then returned the sapphire for a HIS iceq 2x 7970 and for the most part it was a very well put together card that had a terrible ASIC, it failed at minor overclocks and that is just unacceptable for the style fan that came with it.
I've done a lot of research on the 770, and the things that stand out for me are some of the things that are overlooked. Noise/temps/build. The 7970 is a better overclocker no doubt but when the 770 uses less power, runs cooler under load and ovetclocked, and doesn't sound like a lawn mower at Hugh fan speeds, it's the better deal IMO.