thicker barrel on the 590A1s and magazine tubes
From Mossberg 500 Wiki:
Model 500 vs. Model 590 vs. Model 590A1
Mossberg 590 with 20-inch (510 mm) barrel
The primary difference between the Model 500 and Model 590 is in magazine tube design. The Model 500 magazines are closed at the muzzle end, and the barrel is held in place by bolting into a threaded hole at the end of the magazine tube. Model 590 magazines are designed to be opened at the muzzle end, and the barrels fit around the magazine tube and are held on by a nut at the end. The Model 500 magazine facilitates easy barrel changes, as the barrel bolt serves no function other than holding the barrel in place. The Model 590 magazine facilitates easy cleaning and parts replacement, as removing the nut allows removal of the magazine spring and follower.
Parkerized Mossberg 590A1 with M7 Bayonet, upper picatinny rail, and modified standard 590 heat shield
The Model 590 has a plastic trigger guard and safety and a standard barrel. The Model 590A1 has an aluminum trigger guard and safety, and a heavier barrel, intended for military use under extreme conditions and rough handling; the metal trigger guard was added in response to the 3443G materials requirements, and the heavy barrel was added at the request of the Navy. The 590A1 is generally sold through military and law enforcement channels, though in most jurisdictions the 18.5-inch (47 cm) and 20-inch (51 cm) models may be legally purchased by private persons. 590A1s with 14" barrels are Title II firearms, and may be purchased by private persons in NFA-legal states. 
The riot gun versions of the Model 500 (Persuader and Mariner) are available with an 18.5-inch (47 cm) and 20-inch (51 cm) barrel depending on magazine capacity. The 590 is only available with a 20-inch (510 mm) barrel and flush-fit magazine tube. The 590A1 is available with a 14-inch (36 cm), 18.5-inch, or 20-inch (510 mm) barrel. One model sold as Model 590 (catalog item 51663) is technically a 590A1, as it uses the heavy barrel and military trigger group, but unlike models designated 590A1 it is sold on the civilian market.
Unlike Model 500 and 590 shotguns (with the exception of ghost-ring sight 590 models), Model 590A1 shotguns cannot be easily fitted with the factory heat shield, due to the heavy barrel. A heat shield and bayonet lug are required for military 3443G Type I riot shotguns, and some 590A1s are so equipped, but it is not clear if the 590A1 heat shields have ever been offered for sale by Mossberg outside of the military market.