It looks identical to its regular version, Double-Barrel Shotgun, but differs in damage, price and other characteristics. It can only be repaired with Combat Shotguns.
It deals less damage per shot than the Double-Barrel Shotgun but this is made up for with a higher Critical Hit damage. Pa's Fishing Aid, without any perks, in a single Sneak Attack Critical to the head outside of V.A.T.S. deals a whopping 3200 points of damage, which would make it powerful enough to kill just about any enemy in the game in one shot.
This weapon deals the same damage in a non-critical hit as The Terrible Shotgun, but has a much smaller magazine (not clip). Critical hit damage outside of V.A.T.S., however, is much larger due the larger number of pellets.
The name is most likely a reference to the commonly known idiom "shooting fish in a barrel" or a method of fishing known as "shotgun fishing" (throw food into the water, wait for fish to come to the surface, and shoot them with the shotgun).