Defense isn't tipped off to which direction the RB might be running. (as oppose to Shotgun formation)
QB is in a hybrid SG position with eyes already set downfield.
Play action doesn't take long.
Wikipedia advantages
Advantages
The pistol formation can be used in a variety of ways, because the quarterback is closer to the line of scrimmage than a traditional shotgun formation. This allows him to see more easily over the line and make downfield reads. He will also get the ball snapped to him faster, which can alter timing patterns greatly for a preparing defense. The pistol offense can effectively use draw plays, counters, and options using three-wide receiver formations or multiple tight ends combined with a fullback for pass protection. In a pistol formation, handoffs occur 2-3 yards closer to the line of scrimmage than in the shotgun. In the traditional shotgun, run plays are most effectively run to the side opposite the running back, without a cutback to the other side. In the pistol, they can be effectively executed to either side of the QB, opening up more options for the offense. This can make for a more effective running game, but may limit pass efficiency due to quicker recognition of play action by linebackers and defensive backs. This formation works well with dual threat quarterbacks who can both throw and run and is also used when quarterback's mobility has been limited by injury.