At least two of the American cities shown on the map in the Russian sub are misspelled in Russian. Boston is spelled as if it were pronounced "Voston" (using a Russian "B" instead of a "b"), and Philadelphia is missing the initial "Ph".
The sonar depictions of both submarines is incorrect. Sonar screens are a "waterfall" display. The display on many sonars shows multiple Depression/Elevation angles and each band of data in the waterfall shows sound sources in azimuth and integrated over time. The Soviet sonarman would not hear the American submarine behind them as there was no towed sonar.
The Los Angeles-class sub in the movie with designation SNN 710 was called the "Aurora". Its actually name is the "Augusta".
The airplane shown in the shot in the air, from the outside, is a US C-130 Hercules, not a Soviet plane the sailors would be aboard in. The inside is of a much smaller plane, possibly the one shown in the subsequent scene of the sailors disembarking.
Also the second plane shown would not be able to fit all the crewmen of a nuclear submarine.
The smoken is banned in the submarine.
When the Soviet submarine performs its missile targeting routine, the first city locked onto is said to be Washington, but the city indicated by the computer is actually Wilmington, Delaware.
The Admiral addresses a Lieutenant Commander in the situation room as "Lieutenant."