[Alpo]

Pistol:
Pro - can be handled with one hand, leaving other hand free to hold flashlight, turn on room lights, push child back into bedroom, etc.
Pro - is small and can be held close to your body, making someone snatching it away improbable

Con - Short barrel makes it extremely loud inside house.
Con - of the three possible home-defense guns - rifle/pistol/shotgun - the pistol is going to be the weakest of the three. Shoot somebody at living-room range with a 30/06 and he's probably gonna fall down. Shoot somebody at living-room range with a 12 gauge and he's probably gonna fall down. Shoot somebody at living-room range with a 9mm and he MIGHT fall down, or he might stand there and return fire.
Con - pistols are much harder to be accurate with than guns with shoulder stocks - rifles and/or shotguns
Con - pistol normally has greater penetration. Miss the bad guy and go through two sheet-rock interior walls to hit the kid or dog in another room.

Shotgun:
Pro - Much easier to hit your target with a long gun that it is with a pistol
Pro - Buckshot will, pretty much, take down anybody you shoot with it.
Pro - does not have, usually, as much penetration as a pistol, so less likely to go through walls and hit innocents, if you miss the bad guy

Con - loud as hell inside
Con - requires two hands, so holding flashlight or turning on room lights is more difficult
Con - extends two feet in front of you, so while going through doorway, from living room to kitchen, bad guy standing off to side can grab barrel and possibly twist gun out of your hands