If you want to get really picky....
Incorrectly regarded as goofs: The film contains an explicit notice that "certain characters and events have been fictionalized for dramatic purposes", so these changes are not goofs. For instance, the Lovells did not host a party during the Apollo 11 landing; Ken Mattingly was already at Mission Control when the Apollo 13 accident happened, and was not really the person who devised the power-up procedure. There are various other minute contradictions of history and the film is prey to a large number of factual errors due to the large volume of documentary footage/evidence from the actual event. This is not a documentary.
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).
Errors made by characters: 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.
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: 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: In April 1970, Lovell's daughter can be seen holding the Beatles' "Let it Be" album, which wasn't released until May 1970.
Revealing mistakes: When the astronauts are standing in the moving elevator, the reflection in their helmets is of the stationary elevator.
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.
Factual errors: In the launch sequence, we see a countdown that ends with ignition of the first stage engines; we see the Saturn V take off about 20 seconds later, and as Lovell points out, the clock starts counting forward then. In fact the ignition sequence for a Saturn V rocket began 8.9 seconds before liftoff, with the countdown reaching zero at the nominal time of takeoff, not at ignition; the clock would then immediately begin counting upward.
Revealing mistakes: The downward view toward the rocket rising from the pad shows cars in the parking lots. During an actual launch, the pad was completely evacuated and the lots would have been empty.
Continuity: Houston confirms that the BPC (Boost Protective Cover) is cleared before it is jettisoned by Lovell.
Factual errors: Rockets burning hydrogen/oxygen (Saturn V second and third stages), or the hypergolic fuels used on all Service Module and Lunar Module engines and thrusters, have in a vacuum essentially invisible plumes, not the bright white plumes depicted.
Continuity: When Mattingly goes to bed and takes the phone off the hook, the position of the receiver is different when he is woken up.
Factual errors: The actual explosion took place at MET (Mission Elapsed Time) 055:54:53, a full hour before the time shown.
Factual errors: "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." Charlie Duke: "This is Houston. Say again please." Lovell: "Houston, we've had a problem. We've had a main B bus undervolt."
Continuity: Just after the explosion, when Lovell is saying "We've got multiple caution and warnings, Houston," the MET clock (Mission Elapsed Time in hours, minutes, and seconds) is plainly visible reading 091:34:10. When next seen less than a minute later, it has backed up to 056:55:12.
Crew or equipment visible: When Lovell is looking out the window and sees the oxygen escaping, a hand is visible in the bottom left corner of the window.
Continuity: Before Gene Kranz calls for people to "Listen Up People" in mission control, we see Deke Slayton move from the back row to CAPCOM row 3 times.
Factual errors: The seas are the dark parts.
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.
Continuity: Jack Sweigert's "NO" sign is briefly seen on the instrument panel before he actually puts it there.
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.
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.
Crew or equipment visible: When Marilyn Lovell is standing in front of the sliding glass door in her kitchen, a crew member is briefly visible on the left side of the window.
Anachronisms: "Mr. Coffee"-type drip pots weren't in use at the time.
Continuity: A red ashtray and a paper cup in the control center disappear between shots.
Continuity: During the re-entry simulation with Swigert, Fred Haise communicates with Houston after they confirmed radio blackout.
Factual errors: In some cold scenes in the LEM, breath is visible. The warm breath rises, which wouldn't happen in a weightless environment.
Crew or equipment visible: A bearded crew member is visible in the lower right corner of the screen towards the end of the movie, about the time Jack jettisons the service module.
Continuity: Shortly before re-entry, a NASA reading 34,802 feet per second, range to go 26,025 nautical miles," and Gene Kranz has his top button done up and his tie pulled up. Before worker says, " Velocity now and after this shot, Kranz's shirt and tie are undone.
Anachronisms: The television that Blanch Lovell watches the final splashdown on is a Sharp model that was not made until the late 1980s.
Audio/visual unsynchronized: News reporters outside of Lovell's home during landing voices are out of synch with the video (observed on the IMAX version.)
Continuity: At the end of the film, Gene Kranz sits down in his chair and puts his hand to his head. A few seconds later, in the shot showing Ken Mattingly, Kranz can be seen in the background sitting down again in the same manner.
****
Not sure where I got that stuff from now, probably IMDB and maybe some other website(s), then put them all together in roughly chronological order.
The one that made me cringe most at my first viewing was ignition of the Saturn V starting at zero instead of 8.9 seconds earlier. I was a bit surprised at that because of Tom Hanks being a space nut all his life. Worse still, they repeated the mistake in From the Earth to the Moon when they had Dave Scott as a treasured advisor:--
From the Earth to the Moon DVD 5, Behind the Scenes: The Making of...
0:08:50 Dave Scott as technical consultant
0:09:04 Scott driving the Apollo 15 Rover
0:09:10 Ron Howard: He was also kind of inspiring, because at one point he said, "You guys are making a movie and I appreciate that. You want it to be entertaining and exciting, and I know you all have your jobs to do, but here's a chance to actually get a record down, and to show people in a way that documentary footage never really can. And I know there are limits, but isn't that a worthwhile kind of additional goal?" And so it was very inspiring. You'd hear that and you'd just kind of nod and say, "Yes, Commander!"
0:09:39 Brian Grazer: And these astronauts, Dave Scott or Jim Lovell, the way they talk is in such a low-key way that it's such a non-hypey way of communicating, that is completely aberrant to the way we are in Hollywood.
0:09:52 While checking all props, equipment and spacecraft procedures, Scott also coached the actors on the personalities and behaviour of the men of Apollo
0:10:06 Dave Scott: Most of the actors, in fact all of them, were very interested in how Mike Collins did this or did that, or Al Bean did this or did that, so they could portray the individuals accurately. And then when we started shooting, I would sit next to the director and respond to any questions he may have, or tap him on the shoulder and say, "Maybe something like this."
