It's probably weird in the way that you have no players to chat with and you don't get that banter around the table. Also, you're always the centre of attention as a player and as the GM, you're focusing on one person. It could be potentially very hard for both the GM and the player.

1-on-1 works best for mood games. Try breaking out a copy of Wraith: The Oblivion, for instance, or one of the other horror genre games. Those tend to work very nicely with just one central figure, struggling against the Dark.