Factual error: The astronauts are shown taking their suits off before docking, but in real life they were not allowed to do this, in case of sudden cabin depressurization.
Factual error: When Lovell's daughter is complaining that the Beatles have broken up, she slams the album Let It Be into her rack. The scene apparently takes place on the day Lovell learned he would command the mission in October 1969. Let It Be was not released until May 9th, 1970.
Factual error: Technician John Aaron states that the damaged ship will need to use "less amps than this" as he points to a vintage 'Mr. Coffee' coffee-maker on his desk. Mr. Coffee was not introduced until 1972.
Factual error: During the Engine 5 cutoff, they show the abort handle unarmed. However, it is required that the abort handle be armed throughout the entire launch.
Factual error: When the astronauts and their families gather around the television to watch the Apollo 11 moon landing, Neil Armstrong is shown on the TV set coming down the ladder normally, top to bottom of the screen. In the actual historical event, the network's TV feed was upside down, with Armstrong appearing to ascend from bottom to top. The surface of the moon appeared at the top of the screen. Moments later, the network corrected the feed.
Factual error: The fiery rocket plume left behind the Saturn V as it is ascending directly after lift off is far too small and short. In reality the flaming rocket plume was easily 2 to 3 times the diameter of the entire ship and at least 3 to 4 times the length.
Revealing mistake: In the final 1/2 hour there's a scene in which Kevin Bacon is drifting weightlessly in the lunar module, with a roll of duct tape floating nearby. Just during the last couple of seconds of this shot, suddenly he & the roll accelerate to the right. As everyone now knows the zero-g scenes were photographed on-board an aircraft which flies free-fall arcs for up to about 20 seconds before having to level off. The sudden acceleration on the set means that shot was filmed just as the plane was leveling off from a dive.
Factual error: When the cabin temperature drops, an astronaut's breath is visible. His visible breath rises as he exhales. This is an effect of gravity - on earth the water vapor in breath, which is the component that becomes visible in the cold, rises because it is lighter than the surrounding air. Since the astronaut was in a "weightless" environment, his breath should have travelled in a straight path from his mouth into the surrounding atmosphere, rather than rising.
Factual error: In the scenes where all three astronauts are wearing their space suits, they all have a red collar on the helmet and red markings on suits. The LEM pilot (in this case Fred Haise) would have blue markings and a blue collar so that Houston (and others) could distinguish between the Commander and the LEM pilot when they were on the moon.
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.
Factual error: The initial smoke shown coming out of the rocket gimbals during 'ignition sequence start' is not nearly fast or 'enthusiastic' enough. Additionally, there was in fact only a very tiny amount of white smoke/vapor emitted, and it almost instantly developed into a fiery, violent semi-transparent rocket plume. There is just way too much white vapor for too long a period shown in the film.
Factual error: Moments before and during the lift-off of the Saturn V, Ken Mattingly is shown to be watching from a somewhat private grassy field. Wherever he is supposed to be, he is far too close to the launch pad. No unauthorized persons were allowed to be that close, and certainly not in an undisclosed and unsupervised area, not the least of which was for security reasons. The fact that Mattingly is an astronaut would not give him carte blanch to do this, and his training and discipline would prevent his ever attempting doing so in the first place.
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: On several occasions the astronauts address the Capcom as "Andy." None of the Apollo 13 Capcoms were named Andy. Their names were Jack Lousma, Joe Kerwin, Bill Pogue and Vance Brand.
Factual error: During the launch sequence, all nine swingarms on the launch tower are seen retracting, one by one, as the Saturn V reaches full thrust. In real life, only five swingarms would still be attached to the rocket during this phase. These "in-flight" arms would swing away as the rocket lifted off and cleared the launcher.
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.
Continuity mistake: When the Astronauts are in space and are taking their equipment off, look at Kevin Bacon's hands. The gloves he wears under the Space gloves are taken off and the next scene shows him with them on.
Factual error: When the LM separates from the CM just prior to re-entry, what you see in the film is an undocking, rather than a jettison. The difference is that in a jettison the CM's docking probe is pulled out of the CM to make way for parachute deployment, while in an undocking the probe (the triangular shaped thing pointing "forward" of the CM) is left to allow for redocking later. And in case you're wondering how the astronauts transferred between the two while docked with the probe in place, the answer is that they didn't - they dismantled it to create the tunnel. As an interesting aside, in real life the CAPCOM (I think it was Joe Kerwin) gave a go for "undocking" then corrected himself moments later using the phrase "..correction, GO for jettison...)