Why Bb3 and not Bd3 in this position?

I thought having an attack on the h7 square was better. Why place the bishop with the central pawn in the way here?

I think there are three moves here: Bd3, Bb3 and Ba2. My stockfish prefers Ba2.

Bb3 and Ba2 retain the pressure on the d5 pawn and also place some pressure on f7. Bd3 attacks h7. At some point black may play d4 and this will make the a2-g8 diagonal quite attractive for the white bishop. 

So yes, the d5 pawn does get in the way of the bishop but that might not always be the case and, besides, the d5 pawn is a target in its own right. 

So, do you see d5 more as a target for white or an obstacle for black to deal with? Also, if you say obstacle, do you think it is an obstacle more to get the other pieces to the center or is it more of an obstacle because black has to protect the central pawn?

Black cannot castle short because e7 is hanging. One reasonable try is to play Bc6-d7-e6 then castle. Bb3 maintains the pressure on d5 and stops this plan, making black's defensive task a lot harder.

One should elaborate on just how paralyzed black's position really is.  Bb3 and Ba2 are needed to maintain this paralysis.  The c6 bishop can't move at all without dropping d5, because the black queen is preventing mate and thus doesn't count as a defender.  Neither black rook can come into play for that reason.  Both the f6 knight and c6 bishop are needed to defend d5, and are otherwise blocking useful pawn thrusts.

I think Stockfish is preferring 14. Bb3 because Ba2 takes a useful retreat square away from the c3 knight, which may be needed in the event that black plays d4 (which he might, sacrificing a pawn for mobility is certainly warranted in his position).

Flag

Huh?

In this position flagging was better than Bd3 and Bb3...

Flagging wouldn't have been a practical decision. I had 9:07 on the clock and they had 8:45. It obviously wasn't an endgame position. Plus, I don't flag. I don't even play out a game when I blunder usually. I take the loss, learn the from the mistake and move on.

Nice

Bb3 or Bd3 is a choice we have make quite often. You chose Bd3 because you envisioned a future attack along the b1-h7 diagonal. But how realistic is that? It seems to me that Black's kingside is well-protected and you won't have an easy time getting enough pieces over there.Another problem is that once you play Bd3, Black can play Bd7 and c6 and Be6 and his position looks pretty safe. [EDIT or Bd7-e6 and c5 grabbing a lot of the center.] You do have the more active position, but he has that pawn!

This is correct. With open d-file the d5 pawn is on death row.