As Byam is climbing aloft for the second time, he has a bottle of brandy in his hand-- carrying the bottle and attempting to climb almost makes him fall at one point. When he reaches the top of the mast, both hands are empty, and the bottle is nowhere in sight.
During the takeover, loyal crewmen are being shown bayoneted or struck with rifle butts by the mutineers. In fact, the Bounty was taken bloodlessly and Bligh was the only one to physically put up resistance.
When Bligh questions one of the crewmen towards the beginning of the film, mention is made of Dartmoor Prison. Dartmoor Prison was constructed in 1809, twenty years after the Mutiny on the Bounty took place.
When the Pandora runs aground, the prisoners are shown being unshackled below deck and safely released. In reality, four went down with the ship (two still manacled and two struck by a falling gangway.)