The duster provides bonuses of 1 to Charisma and 5 to Small Guns (5 to Guns in Fallout: New Vegas). The hat also provides +1 Perception.
In Fallout 3, the duster has a Damage Resistance of 5, and the hat has a DR of 1. Both can only be repaired by merchants. In Fallout: New Vegas, neither article has a Damage Threshold, so they cannot be repaired by merchants. Repair is only possible through the Jury Rigging perk.
In Fallout: New Vegas, the outfit can be obtained from the town of Primm, inside the Sheriff's office. The duster will disappear after the My Kind of Town quest. Other hats can be found on Freeside thugs from time to time.
Billy Creel asks "What can I do for you, Sheriff?"
Andy Stahl is annoyed but will ask if he "can help, Sheriff?"
Bugs
Playstation 3Xbox 360 The sheriff's duster can still be found occasionally, even after completing the My Kind of Town quest as it sits on the floor in front of the bed with the hat and a wastelander's settlers outfit adjacent to it inside the sheriff's shack. [verified]
Playstation 3Xbox 360 If you are carrying a sheriff's hat, others that are picked up will not appear in the inventory. [verified]
Xbox 360 In Fallout: New Vegas, sometimes, when worn with a set of T-51b power armor in the inventory, the game will act as if the T-51b power armor is equipped, providing the same stat bonuses and damage resistance. This can happen even without power armor training. Whether this is a bug with the T-51b power armor or the sheriff's duster is unclear. It is also unclear if it works in Fallout 3. [verified]