I think that h4 is stronger because it not only stops advancing 'h'pawn but mainly because it also prevents g7-g5.
Points vs Position
basically , white did nothing 'wrong' apart from missing that black had a mate in one which white missed.
Really? At which point? I just rewatched this game and couldn't see it.
LOL indeed cuz it was black mated not white ... or maybe I'm missing sumthin.
Look it's very simple:
1. Black played very strangely.
2. Black gave up material for next to nothing.
3. White was well in front.
4. White missed a mate in 1 and lost.
White
Missed
Blacks
Mate
Perhaps you mis understood?
RainbowRising and Prawn, I think you guys got the games mixed up. I didn't play the game in the first post of this thread.
The game we're discussing is this:
http://www.chess.com/echess/game.html?id=44134200
My sincere apologies for not having clarified that at the beginning.
No need to clear things up, RR and I were talking about the OP.... you know, "On topic". 
LOL Prawn and RAinbow were driving me crazy with their comments, i didn't know what the hell are they talkin about. I was beginning t think that maybe I'm more stoned then I was aware of
.
.
The computer (and I) agree with you that a3 was not a strong move by me. Perhaps a4 was better (that's what the computer likes btw) and I think Be3 is ok too but I still don't like h4. You are a strong blitz/bullet player (at least by my standards) so my guess is that you are much better at attacking chess than I am, so perhaps that is why you see h4 in such a different light than me.
Being strong/weak bullet player doesn't have anything t do with it(mainly because bullet is more about making simple,solid moves fast). h2-h4 isn't an attacking move it's preventing eventual black attack on this side without exposing yourself (black has no immediate option to weaken you by g7-g5).