In the scene just after the S4-B Booster launches the Capsule/Lander combo on the leg toward the moon, one crew-member breaks out a cassette tape and a small cassette player (like a Walkman). Cassette players like this did not exist till the late 70's.
In a scene in the command module, as Aldrin retrieves a pen in free-fall (which is later used to make a broken switch operable), Kubrick's trick from '2001: A Space Odyssey' is employed: the 'weightless' pen is adhered to a sheet of glass to make it appear as though it is floating. As the pen is retrieved, a smudge of glue is visible on the glass.
In new footage, shot for the movie in studio to represent the astronauts' activity on the moon, stars are visible in the black sky. While stars were visible in the sky to the astronauts during the real moon landing, they are not visible in any film the astronauts shot because the cameras' aperture had to be closed down to eliminate glare from the lunar surface. The stars were then too faint to register on the film.