What moves me (as opposed to just impressing me) with SOME fast performances is a sense of 'fury', which Brewtality mentioned. I don't know if I can define it well. Here’s an attempt:
a. I think fury is more likely to happen in passages in minor keys, because it’s connected to darker emotional forces. I think that’s why Barere’s speed playing in the Liszt sonata, Don Juan, Chopin 1st & 4th ballades moves me while his Islamey, Schumann Toccata, Gnomenreigen don’t do so as much.
b. I think the speed somehow has to go hand in hand with a sense of unchecked emotion and/or dark forces. The explosions in the 1st movt of the Appassionata lose their emotional impact if too slow. Or the repeated chord passages in the Dante Sonata. They're meant to be depicting desperate panic at the prospect of Hell, not a stroll through the art gallery. One way (maybe not the only way) is to play them very fast. In fact, none of the performances I've heard (=Cziffra, Berman, Ogden, Jando, Grimaud, Hough) plays them at a tempo that I think realises this (maybe human players can’t do so without losing rhythmic clarity). BTW, I'm not saying every emotional outburst should be maximally fast. The Pathetique 1st movt is often done too fast IMO.
c. The occasional wrong note helps because it adds some dissonance and sounds like the pianist is following the emotion rather than worrying about being squeaky clean. Recall Horowitz' claim (lie?) that he sometimes deliberately played wrong notes to get certain effects. Hamelin is too accurate to generate much fury.
d. With octave passages, an increase in speed is more effective than using one's maximal speed throughout, e.g. the octave passage towards the end of the Liszt sonata.
e. Same goes for fortissimo: listen to passages where you think Cziffra is at fff and he suddenly gets louder, which is better than if he had played his loudest all throughout the passage.
f. Fury seems to be increased if the contiguous slow passages are very slow.
g. The piano is a very good instrument for generating fury on. Many other instruments (guitar, violin, flute, organ, harpsichord) are not as good as vehicles for fury. I think that’s why I don’t feel moved by technical exhibitions on other instruments. As I define the term, fury seems impossible in jazz (Tatum doesn't generate fury), but a good thrash metal band can generate it as well as a good fury merchant at the piano.
Caveat: When talking about speed playing, I am talking about performances in which the performer is not detracting from the piece with too many wrong notes or with lack of musical details like phrasing, articulation, dynamic contrast, etc. If you hear that in a fast performance, it means that the performance is sub-par, not that speed is intrinsically bad.
Yours furiously,
The Cow