So, in the book in 2001, it says the monolith is around Saturn, and in the movie, it says it's around Jupiter. Also, in 2010, they both agree that it's around Jupiter. Why is that?
The book and the film were written at the same time. The original plot was for both to go to Saturn. At the time, they didn't have suitable pictures of Saturn's rings to use for the film. The rings are very much a part of Saturn's image to the public, and they look very different when viewed at the close angles that a moon of Saturn would have. To use Jupiter, they could use a telesope to photograph the planet well enough, and one wouldn't be able to tell the difference in where it was shot.
As for 2010, Authur C. Clark liked the idea of Europa possibly having life, as had been recently identified by the Voyager Missions, that he decided to stage both the book and the movie at Jupiter.
This is a question that doesn't necessarily have a simple answer.
From the Author's Note in 2010:
The novel 2001: A Space Odyssey was written during the years 1964-1968 and was published in July, 1968, shortly after the release of the movie. [...] both projects proceeded simultaneously, with feedback in each direction.
From the Wikipedia article on the film:
The production was unable to develop a convincing rendition of Saturn's rings; hence the switch to Jupiter.
Clarke confirms in his 2010 note that filming Saturn would have been difficult (although Douglas Trumbull later figured out a way to film Saturn in Silent Running, drawing on his experience when doing the special effects for 2001). He also writes:
Stanley Kubrick wisely avoided confusion by setting the third confrontation between Man and Monolith among the moons of Jupiter.
Sir Arthur also indicates in these pages that much of his decision about basing 2010: Odyssey Two among the moons of Jupiter is was a result of the influences of the Voyager flybys of Jupiter, and the striking amount of information we now know about the Jupiter System.
Reading through the excerpts of Arthur C. Clarke's journals in The Lost Worlds of 2001, Stanley Kubrick was extremely enthusiastic about keeping up with current scientific thought in space technology, so it seems feasible that he'd want to show Jupiter and not Saturn on the screen, out of a desire to not be outdated. This answer's point about basing the destination on the photos of what planet was available is feasible, although I think a desire for a simplified narrative may have played a role.
The book and movie of 2001 were developed more or less together, a unique happening to my knowledge, and Clarke and Kubrick made decisions that they felt suited the media of cinema and the written word.
In general, I suspect that the decision to shift the Discovery One's destination to Jupiter for the film was probably based on a number of factors including special effects, accuracy and narrative simplicity. I'll update this if I find out additional information in The Lost Worlds of 2001. (I had an additional book on the making of the film, but that volume fell apart years ago, unfortunately.)
According to Wikipedia (on the film Silent Running), there were also some issues involved in the special effects:
In an interview with Starlog magazine in the late 1970s, Douglas Trumbull...
Trumbull had been involved with creating effects for 2001: A Space Odyssey. Director Stanley Kubrick wanted the Stargate sequence of that film to be centered around Saturn, but there were technical difficulties in getting the special effects for it finished in the limited timeframe. The Saturn idea was scrapped, and Kubrick substituted Jupiter instead. Trumbull developed the sequence after production, and it was recreated for Saturn in Silent Running.