Definitely do not pay 140 for the 7770. But I would definitely go with the 6870 like Azn Cracker said its an all around better card with double the stream processors and twice the memory bus width (256 bit vs 128 bit).
But as also stated neither will max out "most games at 1080p" youll wanna look at probably a 2gb 7850 to start maxing games at that res and even then it will be a select few.
I had a 7870 and with a healthy oc could play Far cry 3 on ultra with 2x msaa but thats at 720p(granted i game on a 40" tv) and i would still dip into the 30's(if thats helpful). My oced 7950 is much better though but without paying at least 300 for a gpu you really wont be able to max out games at 1080. But yea IMHO 6870 is the way to go if 7770 and 6870 are the only options.