I wouldn't say I upgraded from the 7D to the 5Diii, but I have both bodies and therefore can perhaps give a reasonable comparison.
My main photography interest is wildlife/birds, so I have to say that I still prefer to enjoy the crop factor of the 7D. However, the pictures from the 5Diii are better - especially in lower light. The 5Diii provides far less noisy pictures than the 7D across the entire ISO range and I'll happily shoot at 6400ISO on the 5Diii, where I won't be happy at half this figure on the 7D. The resolution and quality of the pictures is simply better.
I find (swapping between the two) the most noticeable difference between the bodies is in the magnification function. The + & - buttons at the top right of the 7D are not there on the 5Diii. Instead, it has a single (left side) button with a magnifying glass icon which is customisable to give you up to 10x magnification at a single press. You then use the top wheel to go in and out of magnification. I constantly try to use the top right (7D configuration) buttons when checking shots on the 5Diii and that's my only annoyance.
There is no question that the 5Diii is a far better camera than the 7D, but then it is twice the price. The AF feels different, but it's still on a par with the 7D. The IQ is better (being FF it will be I suppose). The batteries etc are the same for both bodies.
Speaking only for wildlife, the best way I can quantify it is to say that if the subject is close enough, I'd rather use the 5Diii than the 7D every day of the week and twice on Sundays. Unfortunately however, the subject is rarely close enough and hence my 7D is typically found on my longest lens with a TC in there for good measure too. If all wildlife were tame, I'd throw away the 7D,.... but it's not.
I've no idea if your 'thing' is wildlife, but if it's not, and you don't need the crop factor, then it's a huge upgrade from a 7D without feeling like it's a completely different machine. I believe the 5Diii is one of the top three best bodies Canon currently produce. Hope this helps.