I know this is a bit much, but I hope it helps.
Factual errors: After the party, Lovell holds his thumb in front the gibbous moon. Then, telling Marilyn where to find "her" mountain, he says the Sea of Tranquility is "where the shadow crosses the white part." The terminator was in fact near the Sea of Tranquility on July 20, 1969, but the moon was less than half full; it's depicted in the scene as gibbous, with the terminator on the other side.
Factual errors: When Jim Lovell is standing in his garden looking at the moon (one eye closed) he covers and uncovers the moon (from his perspective) with his thumb. Since the moon was the only light source in this situation, the shadow of his thumb would have to be shading his eye. But the third person perspective shows the thumb's shadow elsewhere.
Factual errors: The seas are the dark parts.
Factual errors: In Houston the moon set that night at about midnight CDT, while the Apollo 11 astronauts were returning to their Lunar Module; hence it would not be visible after the party at the Lovells'.
Anachronisms: NASA's "worm" logo was not developed until 1975.
Anachronisms: A technician is wearing a Rockwell International logo on his coveralls. North American Rockwell became Rockwell International only in 1973 when they acquired Collins Radio.
Factual errors: The launch tower was on the north side of the Saturn V. If Mattingly was watching from east of the pad near the beach, then he would see it on the right.
Anachronisms: In April 1970, Lovell's daughter can be seen holding the Beatles' "Let it Be" album, which wasn't released until May 1970.
chest.
Factual errors: The actual explosion took place at MET (Mission Elapsed Time) 055:54:53, a full hour before the time shown.
Factual errors: The astronauts are shown looking at Mare Tranquilitatis, then crossing from sunlight into shadow, followed by loss of signal, all within seconds. In fact at loss of signal they had been in the moon's shadow for some time and were nowhere near Mare Tranquilitatis.
Factual errors: While passing over Tsiolkovsky crater on the moon's far side, the astronauts also speak of sighting Fra Mauro and Mare Imbrium, both nearly halfway around the moon.
Factual errors: Just after acquisition of signal, Houston tells the astronauts that their speed is "approximately 7,062 feet per second" and their altitude above the moon is 56 nautical miles. That speed is 500 ft/s below lunar escape velocity at that altitude, hence impossible on a free return trajectory. In fact, any free return trajectory symmetrical about the moon-earth line would put them at over 100 nautical miles altitude at acquisition of signal.
Factual errors: A TV scene at Mission Control shows Houston Astros player Jimmy Wynn hitting a home run on 13 April 1970. The Astros were shut out by the Los Angeles Dodgers 2-0 that day. The home run shown was hit 10 June 1967, in a game between Cincinnati and Houston, it was the longest in Crosley field history
Errors made by characters (possibly deliberate errors by the filmmakers): As Neil Armstrong walks on the moon, Walter Cronkite says the Apollo 11 landing is 18 months after the tragic Apollo 1 launchpad fire. It was actually 30 months after.
Anachronisms: "Mr. Coffee"-type drip pots weren't in use at the time.
Anachronisms: The television that Blanch Lovell watches the final splashdown on is a Sharp model that was not made until the late 1980s.
Anachronisms: In the opening sequence with Apollo 1, the crew uses a black keyboard (Block II). The keyboard on Apollo 1 was white (Block I).
Factual errors: In the opening scene, where the Apollo 1 crew is walking across the swing arm, the Service Module is incorrectly shown as a Block II Service Module (grey w/ white areas); Apollo 1, Serial Number 012, was a Block I Spacecraft - its Service Module was all-white, like the toy spacecraft Lovell shows his son later in the movie.
Factual errors: The Apollo 1 fire DID NOT initiate with a switch throw, as this movie depicts. During the Accident Investigation, this was a possibility which was dismissed after a thorough investigation.
Factual errors: The mating of launch vehicle stage components was a much more delicate and deliberate process that this movie depicts. Tours, even for high-level VIPs, would most likely be suspended while such operations were in progress. For the scene where the Third Stage is being lowered onto the Second Stage, a retractable work platform, curved to fit around the Second Stage, would have been in place just below the Upper Field Joint (top of Second Stage). Numerous hard-hatted workers, most on their knees, would have been on the edge of this platform, making sure that the two stages were in correct alignment for mating.
Factual errors: Lovell's Crew was switched from Apollo 14 to Apollo 13 to give Alan Shepard and his crew more training time - not because of any medical issue.
Factual errors: This Movie incorrectly depicts the Apollo 13 Spacecraft and Launch Vehicle as being delivered to the Launch Pad on April 9, 1970 - two (2) days before launch. Much more "pad time" was required, and Apollo 13 was actually delivered to Launch Pad 39A on December 15, 1969.
Factual errors: Notwithstanding that this movie used dramatic license to make a better story, a few points should be clarified, particularly concerning the portrayals of Astronaut Fred Haise and the Grumman Corporation: Fred Haise was considered one of the best and brightest of his group (1966); the scenes where he gives Swigert a dirty look in the Simulator, blames Swigert for the Accident, and nearly screws up the course correction maneuver are not supported by any historical account available to this contributor. Grumman Corporation Personnel worked as hard as anyone else on the team to bring the Apollo 13 Crew home safely; the display of "no can do" attitude shown in the movie is not supported by any historical account available to this contributor. This movie also neglects the contributions of Astronaut Charlie Duke; by the time of the Accident, he was no longer contagious but still feeling unwell. He got out of his sick bed to help devise new Lunar Module Procedures, just as Ken Mattingly is shown doing for the Command Module - in fact, Duke helped devise the course correction maneuver in the Lunar Module Simulator.
Factual errors: Apollo 13 incorrectly shown going around the Moon with Lunar Module (LM) (Aquarius) Landing Legs still folded. These had to be extended before the Descent Propulsion System (DPS) could be fired, which had been done (for the first of three times) many hours before. Once extended, the legs could not be retracted.
Factual errors: In some cold scenes in the LEM, breath is visible. The warm breath rises, which wouldn't happen in a weightless environment.
Incorrectly regarded as goofs: "Houston, we have a problem," is probably the world's most known misquote. After the bang, the conversation was as follows. Swigert: "Okay, Houston, we've had a problem here." Jack Lousma: "This is Houston. Say again please." Lovell: "Houston, we've had a problem. We've had a main B bus undervolt." However, this was a deliberate change suggested by Tom Hanks to better convey the sense of urgency in the scene.
Factual errors: During the trip to the moon, a full moon and a full earth can be seen from the windows of the shuttle. If the spaceship was between the earth and the moon, it would not have been possible to see the "full phases" of the earth and the moon at the same time. One of the two would have to have the shadow dark side facing the spacecraft.
Anachronisms: The controller giving the typhoon prediction for the landing area can be seen holding a full color satellite picture of the region. There were no color satellite pictures at the time - especially not in (near) real time.
Miscellaneous: At approx. 17:47, there is a photo session. The Hasselblad 500 camera used is out of film. It is shown by the red arc on the film magazine. A camera with all film unexposed should view a "metallic" arc, rather than a red one.
Anachronisms: The USS New Orleans (a stand-in for the USS Iwo Jima) is shown with CIWS (Close In Weapons System) that was not in commission until 1977 when tested on the USS Bigelow.
Factual errors: The Mission Clock is not supposed to start until the Saturn V rocket lifts off the launch pad.
Anachronisms: Several scenes show a Lockheed-Martin coffee mug on flight director Krantz's desk. Lockheed Martin was not formed until 1995 with the merger of Lockheed Corp. and Martin Marietta.
Factual errors: During the TV transmission prior to the accident, the crew did in fact play the "Zarathrustra" theme from "2001: A Space Odyssey" as intended. The tape was not switched by Fred Haise as depicted in the film.
Errors in geography: Whilst on the way back to Earth, a view out of a window shows the Moon with around 50% of the far side showing. Only a small percentage of the far side should be visible at that point. When flying over the Moon, Haise looks down at the surface and says "Sea of Tranquillity", when the view out of the window does not show the smooth plains of that area. The scene shown is a mountainous area with a curving rille running though it; it is actually the Apollo 15 landing site at Hadley Rille.
Factual errors: When Apollo 13 is swinging around the moon Fred Haise is looking at the lunar landscape and says "Sea of Tranquility - Neil and Buzz's old stomping grounds". The scene then shifts to show the view out the window and the lunar landscape that is shown is actually the Hadley Rille region, where Apollo 15 would land 2 years later. Besides, on a free return trajectory around the moon the Sea of Tranquility would not be visible while they were swinging around the far side of the moon (it is only visible on the near side of the moon).
Factual errors: Before re-entry, the crew were informed that their course was shallowing. This was stated to be due to them being underweight because they did not have the lunar samples that would have been gathered if they had landed on the Moon. However at this point they still had with them the entire Lunar Module, which would normally have been left behind before the return from lunar orbit, so the total mass returning to Earth at that time was considerably more, not less, than planned.
Factual errors: In the launch sequence, the five F-1 engines do NOT start simultaneously, as depicted. In fact the center, or inboard, engine starts first, followed by 2 more engines and then the last 2 engines just a second or so apart.
Factual errors: All the swingarms connecting the launch vehicle to the tower do not come back one at a time as depicted. There are 9 swingarms, and 4 are removed during the countdown. The last 5 all come back together simultaneously at exactly T=0; the moment of lift-off.