Concerning OP's question and the conclusion the video makes, my experience (after owning 5D2, 6D, 5D3) is the 5D3 is noticably better than 5D2 in terms of noise and ISO performance as ISO increases. The difference in noise is as others have pointed out: less banding, less color noise, and a more "film like" noise pattern.
I can tell you different editors and NR softs will make for slightly different conclusions when comparing the two bodies. I use Capture One v7 here and when comparing in that, 5D3 is noticably better in terms of noise and ISO.
That said, 6D (which I had for a short while and sent back for a 5D3) was even a bit better at noise and high ISO than 5D3. If these were my primary concerns, I'd probably have stayed with 6D. But despite how good the 6D center AF point is in low light, the 5D3 AF system is in a different league altogether, as is build quality, and these swayed me to 5D3 and very happy with it.
And all of that said, I'd take the 6D over the 5D2.