Cole says he doesn't know who Moe Berkus is, but the first time they meet in person, he doesn't behave as though they were just meeting for the first time.
The Apollo 10 launch is depicted as being as the sun is setting on the horizon. It actually launched just after noon. The only night launch of the Apollo program was Apollo 17.
Lance sneers of Stanley Kubrick, "One film and they call him a genius." The reference is obviously to "2001: A Space Odyssey," but Kubrick was already well-established at that time, having previously directed such acclaimed films as "Dr. Strangelove," "Lolita," "Spartacus," "Paths of Glory," etc.
During the Apollo program the firing room at Kennedy Space Center was only responsible for activities leading up to the launch of the rocket. As soon as the rocket left the launch pad responsibility for the mission was handed off to Mission Control in Houston, TX. During the landing the firing room in Florida would not have been actively involved in the mission.
The moon landing site is portrayed as having the earth visible low in the sky behind the LM. In reality due to the location on the Apollo 11 landing site being near the moon's equator, the earth was high in the sky closer to the zenith and wouldn't be visible in this shot.
The movement of dust on the real lunar surface in the film is inaccurate. Due to the lack of air on the moon dust particles travel in perfect parabolic trajectories instead of billowing like they do in air.
The Wolfe's diner was an existing location at the time, but it was not a stand-alone location, it was adjacent to a Ramada Inn which was all located at 900 North Atlantic Avenue Cocoa Beach Florida 32931, which is the current location of the Oceans Landing resort and Racquet club.
Henry Smalls (Ray Romano) would have been clean-shaven, as NASA then had a rule banning facial hair.
A modern red LED digital countdown clock is seen at the back of the room in the mission control center. Such devices did not exist until several years later than depicted. In reality NASA used analogue countdown clocks away from the main TV monitors until well into the 1970s.
Kelly would have been wearing nylons, not bare-legged.
The Vehicle Assembly Building is consistently depicted with an American flag and a NASA logo on the side, which is its current appearance. However, during the Apollo era, there were no decorations on the outside walls. The flag was added in 1976 for the Bicentennial, and the NASA "meatball" logo wasn't added until 1998.
During the party a record stylus is shown and lowered to a record, the bee gees song immediately starts playing from the beginning, but the stylus is shown to be lowered on the end of a track.
When Cole and Kelly stand on "Canaveral Pier" looking toward Kennedy Space Center, they are facing the wrong way. Based on the position of the ocean, they are facing south. The city of Cape Canaveral is south of KSC, which is located west-northwest of Cape Canaveral. so they would have to face completely the opposite direction to see KSC from there.
When Kelly tries to recruit Abby for Project Artemis, Abby repeatedly states she will not work for President Nixon, but in a previous scene Moe tells Kelly that the President hasn't been told about it.