Revealing mistake: When the Russian pod is launched, a wide shot shows the Monolith at top/right with Jupiter in the background and both the Leonov and Discovery at bottom/right with the pod nearby. Just before this shot ends, all three spacecraft vanish, leaving only Jupiter and the Monolith for a very short but obvious moment.
Continuity mistake: When doctors are discussing the medical condition of Dave Bowman's mother, a close-up of the African-American doctor shows a framed newspaper on the wall behind him. It is the front page and has prominent pictures of Bowman and Poole taken during their mission. In the next, wider shot, we see this wall suddenly has nothing on it - the newspaper and frame have vanished. They return when Bowman's mother sits up in bed.