Factual error: When the astronauts pose for their photo shoot, the flag behind them has only 48 stars (a bit hard to tell, since it's rolled up, but the stars are arranged in straight rows parallel to the stripes, which was the arrangement used on the 48-star flag). It's 1970, so it should be a 50-star flag.
Factual error: It is shown that people are casually having conversation next to the giant rocket-ship transporter, known as the 'crawler'. In fact, the crawler is so noisy that no one could possibly have a conversation near it. All technicians who 'drive' the crawler (an eight man crew) and anyone in the vicinity of an operating crawler wear sound-cancelling headphones whenever near it.
Continuity mistake: After Houston reacquires contact with the spacecraft following re-entry, Gene Kranz sits down in his chair, obviously full of emotion. The camera then cuts to a wide shot of Mission Control before panning in on Ken Mattingly. Before the camera zeroes in on Ken, watch the top left corner of the screen and you can see Gene sit down again. (02:09:20)
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.
Factual error: In the movie, Henry tells a White House rep that Blackout lasts for 3 minutes, and if they are not back in 4, then they'll know they didn't survive. This is actually not correct. During the Apollo program, blackout typically lasted about 4 minutes. Also, Apollo 13's re-entry blackout in fact lasted 6 minutes, not 4 as seen in the movie, which was 87 seconds longer than predicted.
Factual error: In the scene where Marilyn Lovell is in her backyard and she looks up at the jet flying over her house (presumably her husband), there is a contrail behind it. The jet is at too low of an altitude to produce a contrail.
Factual error: When astronauts are wearing their spacesuits before launch, we can see NASA's "snake logo" introduced in 1975, when the plot is happening in 1970. (00:28:29)
Visible crew/equipment: During the moon landing daydream sequence, Lovell is shown performing the typical low-gravity moon walk. The thin wires used to facilitate the hopping effect can be seen at the top of the frame, and stand out against the light metal finish of the lunar lander. (01:13:30)
Continuity mistake: When Lovell is writing the gimbal conversions, as Haise says the computer is up, Lovell is clutching the left side of the paperwork and writing with his right hand. In the next frame with the camera closely showing the conversions, Lovell's left hand is now holding the paperwork at the bottom (the left side of the paperwork is now clear of his left hand). (01:01:00)
Factual error: The remaining smoke plume after the Saturn V liftoff is about 80 miles too low. It's apparent that an aircraft flew over the space center to leave a feeble 'movie' plume. Also, when the Apollo 13 crew blasts out of Earth orbit, the ship is pointed straight at the Moon, meaning that once they travelled the 240,000 miles to the Moon's orbit path, the Moon would be several thousand miles to their left.
Continuity mistake: While the first stage of the Saturn V is fuelled by kerosene and LOX and burns with a bright orange flame, the second and third stages are fuelled by hydrogen and LOX, which has an invisible flame. In the movie all three stages burn with bright orange flames.
Continuity mistake: While the flight director (Gene) is talking to a guy about bringing the spacecraft's power down, there is a side shot of Gene saying: "that's the deal?" now, his hands are in front of him (playing with a pen) the very next shot he has his hands by his side and the man next to him has his head in a different position (a very short time to move your arms and head like that).