The single biggest inaccuracy in the movie is that the phases of the Earth and Moon are treated inconsistently throughout it, apparently deliberately so for dramatic effect. At one point, if I recall correctly, both the Earth and Moon are depicted as full, even though the spacecraft is in between them, which would make this optically impossible. At another point, the spacecraft's passage to the far side of the Moon is depicted as passage over the terminator to the dark side, even though this would not have been the case at that time-- and then, later on, the astronauts can see things on the far side, which obviously would not have been possible if the whole far side had been dark.

My impression was that the script was actually far more scrupulously accurate than the special effects, which were tweaked more for storytelling purposes.