When Moe exits the taxi near the start of the movie, he doesn't have the briefcase that he had getting into the cab.
When Moe takes his hat off while entering the dinner that the Professor is hosting, there is a cowlick of hair at the back of his head. He steps into the room with the chessboard and all his hair is now perfectly in place.
The Roman numerals for the newsreel of the All-Americans trip to Japan read 1944; the correct year was 1934.
The slide at the opening says that in 1938 German scientists first split the atom, which is incorrect; that happened in 1919 by Rutherford and others, including Fermi. The German scientists were the first to split the uranium atom.
When Berg first meets Donovan, he is a colonel (eagles on his shoulders), but the sign on his door says "General William Donovan". Now, which should it be? In real life, Donovan was promoted to brigadier general in March 1943 and Berg joined the OSS in August 1943, which would argue for general, but in the movie, Berg shows Donovan his Tokyo film, which Berg screened for intelligence officers in 1942 when Donovan was a colonel. To make things more interesting, we don't if Donovan was one of them and we do know that Berg worked for the OIAA from January 1942 until he joined the OSS in 1943.
Moe Berg is shown in a Red Sox uniform playing at Fenway Park and then later going on his trip to Japan. The film reversed the order of events. Berg's trip to Japan in which he filmed Tokyo occurred in November 1934. He wasn't picked up by the Red Sox until the following season.
As is the case in many WWII movies, the US flag is a 50-star, not a 48-star. Noticed in the meeting room.
When the US baseball team played an exhibition game in Japan, they were greeted by a Japanese band playing "Stars and Stripes Forever". The music and the band are out of sync, which is most noticeable with the musician playing the cymbals. Either 1) recorded music was dubbed over the original band who only pretended to play, or 2) the band was indeed playing, but the audio track was added out of sync.
As a general, Donovan's star would be on the center of his shoulder strap, not at the end like lower ranks.
The action in Italy is just before the Allies liberate Rome, which happened on June 5th, 1944--early summer. But the mission team are dressed for winter, the trees are all bare, and it looks very wet and cold.
A scene from 1943 has a character saying that his parents were sent to Auschwitz. The speaker could not have known of his parents' destination and would not have been aware of the existence of that camp. Likewise, those listening to the speaker would not have known either and yet understand what being sent to Auschwitz meant.
A C-45H Expeditor brings Donovan to Rome to meet with Berg has the marking USAF under the wing. The USAF was not organized into a separate service until 1947, it was still the US Army Air Force in 1944.