When the Americans are flying to Poland, the plane shown in flight for a few seconds has two engines. When it lands, it has four engines.
During the 2nd game (1st one shown), the Black Rook completely disappeared between shots.
The American flag is displayed improperly on the wall during the chess match. According to US flag etiquette, the blue star field should always be displayed in the upper left hand corner when the flag is hung on a wall.
The sign on the table during the matches reads "USSR" in English. Of course, it would say CCCP in Russian.
During Game 4, there is an fifteen minute intermission. Chess matches would not include such a break because the break could easily be used for other people to illegally interfere and feed analysis to the players.
No GM would ever refer to an exchange of a Bishop for a Knight as a sacrifice, it is an exchange.
In the TDS opinion statements at the end, the "USSR" is referred to as "USRR [sic]."
A draw is offered after playing a move, the first draw is offered without a move being played.