The GTX 770 is slightly faster than the 280x, but only in certain games. From benchmarks, it looks like the 770 scales at high resolutions better than the 280x. It also looks like the 770 supports AA better. If it matters, the GTX 770 consumes about 10-20 less watts than the R9 280x. I can't find any temperature benchmarks for the 770/280x DirectCUII. Even though it performs slightly worse, I would buy the 280x because it has better price/performance.
EDIT: It's really hard to find these two compared together in the same benchmark. One website has the Asus 280x at 73C under load, and the reference GTX 770 at 80C. Based off of that, I would assume the Asus GTX 770 runs slightly cooler than the 280x. Techpowerup has the Asus GTX 770 at 76C under load.