Deliberate mistake: Near the end of the movie, there is a shot of the "Iwo Jima" with the two recovery helicopters taking off. For a moment, you can see the number 11 on the inboard side of the ship's island, which identifies it as the USS New Orleans (LPH-11), a sister of the Iwo Jima. Admittedly, it was the best the producers could do, since by the time the movie was made, the Iwo Jima (LPH-2) had been decommissioned.
Deliberate mistake: During the scene where the astronauts are building the makeshift CO2 scrubbers, the camera zooms out of the lunar module through the window, and we can see that the interior of the lunar module is oriented incorrectly compared to the exterior. The tunnel leading to the command module is on the top in the interior shot, but on the left in the exterior shot.