Continuity mistake: The Bismarck is sighted by a British agent in southern Norway, sailing out of the Baltic into the North Sea - east to west. But the view through the agent's binoculars shows the ship sailing from right to left on the screen - which would be sailing west to east from the point of view of an observer on the Norwegian coast.
Factual error: Here's a big historical mistake. The character of German Admiral Lütjens is depicted overall in this film as a wild-eyed Nazi fanatic. In real life, he was distinctly anti-Nazi, vehemently protested the anti-Semitic actions of Hitler's regime, and was himself subject to intense Nazi scrutiny as he was a quarter Jewish and his wife was half Jewish. He was one of many German naval officers who fought only for their country, not Hitler.
Continuity mistake: Near the beginning a reconnaissance Spitfire overflys a Norwegian fjord, taking high altitude photos of the ship. The movie cuts to the photo lab, where a technician develops the photo. The fjord and the ship can be seen in the tray. However when the technician and his assistant remove the print and study it, you can see (through the back) that this is not the aerial shot just seen, but is a picture of the ship from sea level as viewed from another ship.
Other mistake: When the planes from the carrier Ark Royal first spot a ship through a hole in the cloud cover we see a ship sailing along. (The Sheffield, which they mistakenly attack). Later, they supposedly spot the Bismarck, yet we see exactly the same ship sailing along through the same hole in cloud cover.
Factual error: The Director of Operations is handed a fuel state for KGV and Rodney, and is advised they are 140 miles astern of Bismarck. He asks if they steam at full speed can they catch her. Since Bismarck can steam at 30 knots, and the British ships at 28 and 23 knots respectively, it was a question that shouldn't have even been broached.