Can someone explain this to me real quick, because now I'm confused. If those cards only have DDR3, and less streaming processors, how do they outperform cards with higher specs?
To directly answer your question, I don't know.
But, I think it is an academic point.
Graphics card engineers select the components best suited to achieve a desired price/performance point.
They will mix and match ram speed, bandwidth, clock rates, numbers of shaders, etc. The bottom line is performance which you can get some indication of by benchmarks.
In this case, one of the big differences is that the 7xxx cards use a 28nm manufacturing process which is more power efficient and cheaper to make than the 6xxx series.
If you have a particular favored game, then cards in the same price bracket may perform differently, and the similarly priced nvidia card may well do better.
I got a 7750 on Black Friday and can play BO2 at 1280x1024 at full 60FPS at near max with details turned up to quality in the CC panel. Looks great, card doesn't need a powerful PSU at all, not a double slot card, so fits in just about any PC. It should handle any older games at near max.
Here you can see the 7750 vs the 6670 and how much better it performs and also some FPS you would expect in some of the games you listed.