Am not an expert but... that doesn't show loudness difference to me. That shows dynamic range difference to me (even the Youtube description says they actually normalized the volume). I can hear the difference between the two version and how distorted the retail version is compared to the more clean sound of the GH3 version but it's not simply up to loudness.

Honestly maybe loudness war is a wrong term? Should be dynamic range war or compression war.

The final mastering/mixing has an avg db level they target. The more compressed mixes tend to have much louder avg db. Some of these modern "loud" mixes target -3db whereas the previous ones could be -10db or something. Played at an identical volume, the more compressed track will sound louder due to this target. Loudness war is an apt name. Just look at the sample waveforms on the wikipedia page:

* https://en.wikipedia.org/wiki/Loudness_war

The retail track was normalized. You can tell because the GH3 version has peaks that extend far past where the retail version cuts off.

I had trouble figuring that out at first because I thought that one of the tracks would be retail and the other would be GH3. What the video actually shows is the left and right channels of the edited together retail and GH3 tracks.

Well you get things louder but you sacrifice dynamic range for it. If you do not normalise it.

The psychoacoustic effect of compression and loss of dynamic range is a perceived increase in loudness.

Not only is Rick Rubin's mix brickwalled, it's also riddled with clipping because he didn't even brickwall it correctly. It's one of very few albums I find unlistenable merely because of the mixing.