Approval of God: In a way. Stanley Kubrick was offered the film but had zero interest in doing it. After he was hired, Peter Hyams reached out to Kubrick to make sure he was all right with a sequel to 2001 being made. Kubrick said he didn't have a problem with it, and encouraged Hyams to pursue his own vision of the story instead of feeling obligated to replicate the style of the first film.
Fake Nationality: Messed around with. Dr. Chandra is supposed to be Dr. Sivasubramanian Chandrasegarampillai, originally from Madras, but a naturalized US citizen at the University of Illinois Urbana-Champaign. When they cast the Russian-Jewish American Bob Balaban in the role, they dropped any reference to his Indian heritage (and changed his affiliation to the University of Chicago, to boot).
Dmitri Moiseivich was played by an American actor, and Captain Tanya Kirbuk by a third-generation Russian immigrant to the UK (see Fake Russian below). Kirbuk's name is, of course, Kubrickspelled backwards without the "c", but it's probably intended to resemble Ukrainian names that end in "-uk". The other Soviet characters were played by four Russians, one Latvian, one Ukrainian and one Czech.
Fake Russian: Dana Elcar as Moiseievich has the most trouble with his accent; his Russian radio messages are dubbed. Helen Mirren's grandfather was Russian (their family was originally named Mironov) and she can pull off the accent, even though she isn't fluent in the language. The rest of the actors playing the Leonov crew are variously actually from the Soviet Union or Czech.
Life Imitates Art: Around the year 2010, scientists began noticing the color of Jupiter's atmosphere is changing, just not as fast as it happens in the film.
The Other Darrin: Roy Scheider as Heywood Floyd, replacing William Sylvester from 2001. (A still of Sylvester as Floyd, touching the monolith on the Moon, can be seen in the recap sequence in the beginning.) Averted by Keir Dullea and Douglas Rain, who reprise their roles as Bowman and HAL.
The space pod that appears in Watto's junkyard in Star Wars Episode I: The Phantom Menace may also have come from 2010's Discovery pod bay, but this has not been confirmed.
Averted for the film itself, though. Specifically to avoid this trope, Kubrick had all props and sets from 2001 destroyed. Everything had to be rebuilt from the ground up using references from the original film.
Reality Subtext: In the film, the Soviet minister explained to Floyd that their ship's name was changed from Titov to Leonov because Titov "fell out of favor". In real life, Gherman Titov was a bit of a rebel who was busted multiple times for drunk driving, fraternizing with women, and other misdemeanors. The Soviet Union considered officially kicking him out of the program more than once, and likely would have eventually if not for Yuri Gagarin's death in 1968.
Science Marches On: This was Clarke's motivation to keep writing sequels to 2001. The plot of 2010 was inspired by the Voyager probes' flybys of Jupiter, especially the possibility of life under Europa's icy crust, and he wrote 2061 in response to the 1986 observations of Halley's Comet.
This made Stanley Kubrick's version of the plot, moving the mission in 2001 from Saturn to Jupiter, very fortuitous. The first novel sent Discovery to Saturn, with the Monolith found in the vicinity of Iapetus (spelled "Japetus" in the novel, a common British spelling for the moon). The most interesting feature on Iapetus is a ridge that gives it a walnut-like shape. The real-life idea that Europa is now considered more likely to harbor life than Mars makes this change downright fateful!
After Clarke died, the Cassini probe showed that it would now be possible to do a story about a monolith on Iapetus protecting emerging life on Enceladus, since it has even better evidence of liquid water under its crust than Europa does.
Shout-Out: The prop designers for Babylon 5 used the design of the Leonov as inspiration for the Omega Destroyers used by the Earthers on that show. Compare the rotating gravity sections of the two ships, they have a largely identical profile. The designer of the digital prop on B5 assumed that someone would catch his obvious joke and have him fix it, but nobody noticed.
What's odd is that the film version of 2001 gave HAL's creator's name as "Mr Langley", rather than "Dr Chandra". You have to wonder why they didn't simply rename the character Langley when they cast a white guy...