Captain Bligh is shown keel hauling a sailor for a minor offense. The British Navy discontinued the practice around 1720, decades before the Bounty sailed. The French and Dutch navies discontinued keel hauling in 1750, 37 years before the Bounty's voyage.
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 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.)
Bligh and Christian express animosity toward each other at the beginning of the voyage. In actuality, not only were the two men close friends prior to the mutiny, their families had also been close for years.
Bligh did not sail on the Pandora in the search for the mutineers. Captain Edward Edwards commanded that mission. The only Bounty crewman with Edwards was Thomas Hayward, who was one of the men cast adrift with Bligh.