Yes and no. Thanks for asking.
The original Auto Ordinance company had made up thousands of parts for the Thompson sub machine gun expecting a lot of sales. But the gun never took off. Originally they thought the military would want them as Thompson envisioned the gun as a "trench broom" to sweep the trenches clean.
Auto Ordinance tried to sell it to police departments with limited success. They even made up ads of western ranchers defending their herds with a Thompson gun. Well, some members of the public saw a use for them and either bought or stole the guns for gang related crimes.
Fast forward to WWII and the need for large amounts of weapons to supply fresh troops. Auto Ordinance received orders for "military" style guns. The original barrels with the intricate cooling fins were replaced with smooth finished ones. The Cutts Compensator, a device on the end of the barrel to divert gasses upward to help hold the muzzle down while firing was eliminated. The high quality gun bluing was replaced with military Parkerized finish, which is a flat non-reflecting rust inhibitor. Additionally, the rate of fire was adjusted downward for the military guns so they wouldn't use ammo so quickly. Being original receivers, the WWII gun will accept drum magazines, but they weren't typical issued to troops. Box magazines are typically in 20 and 30 round sizes.
The rest of the parts were made from original supplies of Auto Ordinance manufacturer. By the way, there are still supplies of the original parts around today.
The reproduction Thompson submachine gun available for purchase today is essentially the same, but it has a longer (legal) barrel and modern finishes.
I bought and owned an Thompson 1928 AC overstamp submachine gun in the 1970's. Yes it is legal. You have to purchase a special stamp from the BATF and register the gun. My Thompson was made in 1928 but refurbished for WWII with the items mentioned above.
The first submarine I served on had Thompsons in our small arms locker for topside defense. These were later replaced with M-14 rifles and later yet by riot shotguns.
So, yes they were the same guns, but modified for combat use.
I have some books around here somewhere about the development of the gun, but I'm doing this from memory.
WOW, that brings back a whole flood of memories. Thanks again for asking.