Disclaimer: my experience with 6D is limited to a single four-day rental because one of our 7D bodies was in for repair. I shot two charity walks with it, something we enjoy doing but absolute image quality is simply not a factor (getting the images culled/cropped/delivered fairly fast is). I didn't notice any secret sauce with that camera. I'd rented a 5D prior for a charity event, but wasn't experienced enough to really look for the sauce.
OMG, yes, there's some sort of secret sauce in FF. I feel like almost every EF (as in "every Canon lens that's not an EF-S") lens just makes more sense on FF, and the images are just more WOW across the board. This goes for 5D3, 5DsR, and 1Dx, having owned 40D, 7D, 1D3, and now 5D3, 5DsR, and 1Dx, while also having rented 5D, 5D2, 6D, and 7D2.
Amongst other things, years ago I had a nifty fifty. I used it on my 1D3, and once I finally went back to TDP's review and realized that it was best if I stopped down to f/2.8, I discovered that it was a decent lens, but felt the focal length just wasn't all that amazing (and with a 16-35/2.8, 70-200/2.8IS, and later a 24-70/2.8, it didn't serve a purpose, and got sold). Now that I've gone FF, I've picked up the 50/1.2 and wowsers. Sure, I can shoot with that lens for a while and just crank out magic all day long. I had a blast at a baby shower shooting the kiddos playing with bubbles and such.