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. In April Ringo was still recording drum tracks, not even possible for an advance copy to get out.
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: 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: 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: 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: 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: In the scene right after the Apollo 11 landing, where Lovell is in the garden with his wife, he is playing with his thumb and the waxing moon. He covers the moon with his thumb repeatedly.Usually the diameter of the moon disk is about half of the diameter of the thumbnail on an outstretched arm of an average adult. Since Lovell's arm is not fully stretched the moon should appear even smaller. In this scene we see a moon approximately of the size of the whole thumbnail. In relation to Lovell's thumbnail that moon is far too large. (00:05:05)helmholtz42
Factual error: The paint scheme shown on the Apollo 13 rocket is incorrect: The paint scheme shown in the movie is what was used on the development versions of the Saturn 1-C first stage, and featured a large black band around the middle. Technicians found it much too hot to work inside the pre-launch first stage from the heat generated from this paint scheme. Subsequently, the paint scheme was much simplified to a near all-white version for the first stage beginning with Apollo 4. All subsequent ship-stacks (including Apollo 13) featured this newer paint scheme.
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.
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.