FF Sensor Question

I currently own (probably) too many cameras, the Sony RX-1 being one of them, as well as an Alpha A65 DSLR. The RX-1 is a PHENOMENAL (FF, P&S) camera. However, I would like the versatility a FF DSLR (supposedly) offers (like a 6D). The tentative plan would be to sell off my A65 AND RX-1, for a Canon 6D. At the risk of revealing my true inexperience, are the benefits of a FF sensor simply a larger frame? Or does the sensor size also impact image quality? Let's say you have 2 cameras, one APS-C, one FF, with the exact same Zeiss glass, which takes better pictures? Or is it a wash?