Revealing mistake: When the shuttle pod makes its first pass on the port side of the space dock (starboard of the shuttle) a piece of the earth shows over the last strut as the pod passes (blu-ray and Netflix version).
Continuity mistake: After Ilia is abducted by the V'Ger plasma energy beam on the bridge, her tricorder falls on the chair. The chair is facing a different angle than it was before. Also, you can see the white tips of a pair of shoes from someone standing next to the chair on the left. Presumably, this is supposed to be Kirk, but he is too far away to see his feet in that shot.