Upload & Sell: On
Simple answer...yes...you will see a difference! Corners will be a bit sharper, even with my older lenses; a 70-200 f4L IS and 300 f4L IS, when pixel peeping or printing larger then A3 there is a slight improvement. I've both the II and III; got a refurb from Canon, if you are thinking of getting any of the new Mark II telephotos don't hesitate getting the III. Peter/PetKal and others that shoot the big whites do see improvement in IQ and AF tracking with the III.
If you don't print larger then A3 and mostly just view your images via screen/web, I'm not sure the extra buckaroos are needed spent. I shoot both film and crop digital, with my EOS-3's and Fuji Velvia 50 high-res scans the corners are better, as for FF digital I'd expect the same...
Your 5D3 and 70-200 II are the best Canon gives us, maybe rent the III before you buy, I'm guessing LensRentals might have one you could test!