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: 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 Lovell's daughter is complaining that the Beatles have broken up, she slams the album Let It Be into her rack. The scene takes place on the day of the initial explosion aboard Apollo 13, April 13 1970 - immediately prior to the Lovell family attending the screening of a television broadcast from the spacecraft. Let It Be was not released as an album until May 9th, 1970.
Other mistake: When he finds out he is being bumped to the Apollo 13 crew, Lovell is giving a tour at the vehicle assembly building at the Kennedy Space Center in Florida. In the next scene, he runs into his house, which seems to be in Houston - over 1000 miles away - to announce that he's going to the Moon in April, then races off to get back to training. It seems like an awfully long commute.
Other mistake: Mission Control in Houston calls up "B.P.C. Clear", meaning that the 'Boost Protective Cover' has been safely jettisoned during ascent. However, the call comes on screen before it is shown being jettisoned.
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.
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. In reality, Mattingly was in Houston at Mission Control at the time. Otherwise it is a nice shot.
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.
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").
Factual error: The White Team, lead by Gene Kranz did not handle the launch of Apollo 13. This was done by Milt Windler's Maroon Team.
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: 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.
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.
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.