There is only one distinction other than the barrels were made thinner on so called shotguns, the one distinction that earns the shotgun name, putting some form of choke to the inside of the barrel giving that gun barrel a specific job to do, shoot shot only. Once you put choke inside of a gun barrel you then have determined a specific use for that gun barrel, the purpose of that choke is to shoot shot, hence shotgun.

 The purpose of a shotgun having choke is to extend the distance at which you can effectively shoot shot and cleanly break targets or harvest upland game.

 Note: As the addition of rifling to a musket bore extended the accurate distance of a single projectile clearly defining (a rifle), choke boring of a musket bore extended the distance of multiple projectiles (shot) clearly defining (a shotgun).

Black Powder Shotguns / Choke Bore.
 Fred Kimble in 1868 invented the choke-bore shotgun barrel that would forever seperate muskets from shotguns and revolutionize the shotgun with one of the greatest of all barrel inventions.

 Kimble started experimenting with musket barrels left over from the Civil War, as they were heavy and would stand boring. After repeated attempts he finally came up with a 6-bore that would drop birds at 80 yards. This gun would shoot #3 shot through a one-inch board at 40 yards.

 Kimble began challenging anyone and everyone at live pigeons with his 6-bore, 36-inch barrel, muzzle loader. The inferior guns of Kimble's competitors were no match for Kimble's choke-bored single shot shotgun. He won match after match.

 Soon, everyone wanted a choked gun, and Kimble began to procure regulation double guns and bore them to his new shotgun bore dimensions.

 Fred Kimble died in 1941 at the age of 95 and was inducted into the Trapshooting Hall of Fame in 1969, a well-deserved honor.

Black Powder Shotguns / Determining Gauges.
 Determining gauge was simply how many lead round ball of its bore diameter would it take to consume a pound of lead. This was a simplistic approach and rather effective as shotguns could still be loaded with round ball and used for a wide variety of tasks.
 Example: A 50 caliber lead round ball weighs approximately 184 grains at true diameter, there are 7,000 grains in a pound, 7,000 divided by 184 = 38.04. The 50 caliber would be approximately a 38 gauge shotgun.