A comment from the original YouTube video...
I think a lot of the difference is in the action of the keys. The way they feel when you play them. I've been playing a €5000 Hyundai piano for 20 years. The first time I played a €50,000 Steinway in the showroom I was impressed most by how light and accurate they keys are. I was able to play the fast passage in the middle of Mozart's Turkish March at full speed with just a light touch of my fingers and hardly any wrist movement at all, whereas with my Hyundai playing the same notes at that speed needs significant muscle involvement and usually results in slight soreness in my wrists and forearms by the 3rd or 4th recital.
Apparently the keys on an expensive piano are individually balanced and adjusted by hand - and I imagine with 88 keys that process takes some time, and the salary of the craftsman is not cheap and gets reflected in the final price.
In this video the audio difference was clear. The cheap pianos sound tinny and muddy. The expensive pianos sound full-bodied and the individual notes were clear. The sound is probably proportionate to the size of the strings/piano (the last piano appears to be at least twice the sound of the first one). And probably much of the clarity is due to the better hammer action that makes it easier to hit the notes precisely instead of mashing them.