Revealing mistake: When Danilov and Kruschev are in the war room, when they think Zaitsev is dead, pay attention to the maps on the walls. These are modern US and British defence maps that weren't available until starting about 1960. The clue is the the numbers on the squares. These are flight altitudes clearances for jets. Also, you can see that the maps are in Roman letters. Soviet maps during WWII were in Cyrillic and they looked differently.
Revealing mistake: Midway through the movie, there is a long-shot of several characters walking outside between two gutted factory buildings. The camera pulls back, zeros in on the base of a huge chimney and pans upward to the very top - where two men in black and white checkerboard jackets are working.