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.
Continuity mistake: Early in the movie when Thomas falls overboard, John Smith ties a rope around his waist (00:02:56) and jumps in after him. Thomas is very far away from the ship (00:03:12). But when the men pull John and Thomas up, John is holding on to the rope with one hand and the end is left dangling (00:03:36). One of the men who pulled them up then coils the rope with a mere two coils (00:03:48), not nearly long enough to have reached Thomas while he was overboard. In the same scene, when we are viewing the men pulling John and Thomas up from above (00:03:38) there is no extra rope at their feet as there should have been for such a long rope.
Factual error: At a very early point in the film, the King says something along the lines of "....and these are your playmates". In the background is a horde of 17th century ladies boating on the Achille Duchenne water parterre at Blenheim Palace. The palace wasn't built until the late 18th century, and the parterre was not designed until 1925.