Classic cowboy requires either a hammer double or a lever action shotgun, among a list of other things. Other than that, you can use either one in all other categories.
The Stoeger is faster, as you don't have any external hammers to thumb back.
The IAC (Norinco's new name) has more style points.
Although either can be used straight out of the box, both can be made better by some action work. The TTN double is actually the hammer shotgun of choice in CAS...but you'll see some of the IAC also. Lots of cowboys shoot the Stoeger (including me and the wife).