Continuity mistake: In "2001", Bowman steals aboard Discovery through the emergency airlock. In the airlock's alcove is a spare spacesuit, green in color. He takes and uses the green helmet to cross the decompressed pod bay to access HAL's memory. After disabling HAL, Dave wore his red suit and helmet to leave Discovery for the last time, and would have had no reason to move the green suit before leaving, and no one else is aboard to have done so. In this film, we see the same alcove is entirely empty - the green suit has vanished.
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.
Continuity mistake: Dr. Chandra is in the logic circuits room reactivating HAL. He types words into a computer on the wall opposite the logic circuit panel (panel with the clear plexiglass blocks). In the movie 2001 there was no computer screen there; when Bowman is deactivating HAL you can see the flat metal wall with the red grid on the wall opposite the logic circuits.