Factual error: Sergeant Pierson has an M1928 Thompson with a 50 drum magazine throughout the game's campaign. By 1944, the M1/M1A1 Thompson would've been standardized for most soldiers and Marines. And even if he didn't get the M1 variants, he should be using 30 round box magazines instead of the drums. The box/stick magazines were more reliable and lighter than the drums, and would've completely replaced them by 1944.
Factual error: During the beach invasion cut scene, several soldiers have helmet bands with photos and cigarette packets secured to it. In reality, this was not done during World War II; would've been done extensively during the Vietnam War, but not in World War II.