When Floyd drinks out of a straw in zero-g, the liquid moves back down.
One of the flight crew leans on a chair (in zero-g) to talk to Floyd.
The Aries lands with its cockpit windows facing upward, so the pilots shouldn't be able to see the Earth moving up past the windows. (Perhaps the windows have some kind of transparent display overlay?)
When the Earth is seen from the moonbase at Clavius and the Monolith dig site in Tycho, it's oriented with north pointing upwards. However, Clavius and Tycho are in the Moon's southern hemisphere, so the Earth should have been upside-down (and angled proportionately to the viewer's lunar latitude).
In some scenes where Frank or Dave are jogging around the center ring, you can tell they are not quite at the "bottom" of the set and thus are at a slight angle where they would typically be at one. This is when the camera itself is occupying that spot.
The people wearing the shoes that stick to the floor try to walk the way they would in zero G, but in reality they would be slightly fighting the inertia of their upper bodies wanting to stay behind. Instead they just walk as if through glue.
"I'm sorry, Dave. I'm afraid I can't let you do that."
"My god, it's full of stars!" — This line appears in 2001 the book, but not in the movie. Nevertheless, in 2010the movie, it's claimed Bowman said this before entering the Star Gate.
Any time jaunty classical music is used in a space setting, particularly Johann Strauss Jr.'s Blue Danube waltz.
Doing It for the Art: Stanley Kubrick and Arthur C. Clarke spent enormous efforts into making everything as realistic as possible. The earth moving equipment seen on the Moon would actually work on the real Moon. Quite a few experts from NASA and IBM were asked to help design the sets.
Clarke published a few lines from his diary from pre-production in the introduction of a re-issue of the novel. They include "rang Isaac Asimov to ask him about the biochemistry of turning herbivores into carnivores." (Asimov, besides writing science fiction, was a professor of biochemistry.) And they never even did anything with that...
Kubrick required the compositing work to be done by a team of British animators painting traveling mattes by hand frame-by-frame to mask out each element, rather than using bluescreen. When production ended, most of them signed onto Yellow Submarine in order to work on something colorful after spending two years painting little black blobs.
Instead of storyboarding the docking sequence, multiple model sequences were shot so Kubrick could edit them down.
Cast the Expert: After failing to find a British actor who could play the Mission Control CapCom, Kubrick hired a real U.S. Air Force air traffic controller stationed in Britain. The interviewer from the BBC was also played by a real BBC newsreader.
Fake Russian: Brit Leonard Rossiter as Dr. Smyslov, the guy who grills Floyd about just what's going on at Clavius.
Flip Flop of God: What exactly the orbital platforms are for. Originally they were intended to be nuclear delivery systems, but this was later retconned to leave their purpose ambiguous.
Prop Recycling: Deliberately averted. Kubrick had all the sets, special effects models, and design notes destroyed after filming was complete, to prevent them being reused in low-budget B-movies. The production crew for 2010 had to rebuild everything by examining the film itself, frame-by-frame. A deliberate case of No Plans, No Prototype, No Backup.
It didn't work. Several models (rebuilt or maybe the same film clip) have been used. Space: 1999 used the same rocket landing site on the Moon.
Shrug of God: Certain ambiguous or unrealistic elements have been shrugged off by Kubrick and Clarke, such as the true meaning of the Monolith or how HAL was able to read lips from the side.
What Could Have Been: Kubrick had allegedly asked Osamu Tezuka to work as a production designer for the film, but sadly, The God of Manga was far too busy with his own projects to oblige.
Also worthy of note is that Kubrick approached the rock group Pink Floyd to do the music to the film (as well as the later Clockwork Orange), but they declined. Roger Waters later said not scoring 2001 was one of his biggest regrets. (Supposedly "Echoes" syncs up to the third act of 2001, try it out.)
In turn Waters asked Kubrick to use a sample of the "My mind's going, Dave" dialogue on the beginning of the track "Perfect Sense (Part I) on Waters' 1992 solo album, Amused To Deathnote Roger meant to use it as a Take That to ex-bandmate-turned-bandleader David Gilmour, with whom he was feuding five years previous for the rights to the Pink Floyd name; Kubrick refused to let him. Waters instead left a backwards Take That to Kubrick in place of where the ''2001'' dialogue was to be on the album.
Hal is activated on January 12, 1992. The day, now passed, was not particularly notable.
Notable, but not plot-relevant. The scene with the primates is obviously not 2001, but the title card for the Discovery mentions that it is 18 months later. This places the trip to the moon (and discovery of the monolith) in 1999.
Some say the discovery of the monolith happened in 2001, which sets up the odyssey.