Is there a reason why 3...Kf8 is better than 3...Kh8?

What's wrong with 3...Kh8?

I also chose Kh8 over f8

Consider the game continuation. 5. Qa4 in the continuation fails because of Rxd3, renewing the threat to the d1 square. If the rook takes the the black queen, then black responds with 6... Raxd1+ (or Rdxd1+; it doesn't matter) 7. Qxd1 Rxd1#. If the white queen takes the rook, then it's again mate: 6... Qd2+ Kf1 7. Qxd1+ Qxd1 8. Rxd1#. I hope this will also answer mylaguna's question above.

But if you play 3... Kh8, then you can't play Rd3 because of Qe8#. You have nothing better to do after 5. Qa4 than to play Qxc4, which will be followed by 6. Qxa1 Bxd1 7. Qxd1 Qxe7 and the attack has vanished in a cloud of exchanges that leave both parties with a queen, rook, and six pawns. Black's pawns are more advanced and the white king is stuck in the center, but this is only good for a slight plus.