Factual error: Reference is made to the king as George II, and Captain Sawyer as one of "Nelson's own." However, Nelson was only about two years old when King George II died. The Napoleonic Wars were during the reign of King George III.
Continuity mistake: Acting Lieutenant Hornblower is confined to the "hole", a slimy, rat-infested cell where he can neither lie down nor stand. He is confined long enough for Hunter's leg wound to heal sufficiently to regain his mobility, yet, when he emerges, he is clean-shaven.
Plot hole: As the Spanish fleet looms out of the fog, Hornblower is able to identify individual ships, a remarkable feat since every ship is wearing the same paint job.
Factual error: Hornblower's plan is to sail Le Reve south from Gibraltar to give him ample sea room before he heads for England. Generally, the prevailing wind through the Straits is westerly, unless there is a Levanter, yet Le Reve is shown on a port tack when it should be on a broad reach. The fog that comes up would not exist in a gale-force Levanter.
Continuity mistake: When the Spanish demand Hornblower surrender his ship, the battleship fires two shots.the first shatters the topsail spar at the jeers, bringing it down, and the second.shatters the once again undamaged topsail spar outboard of the jeers, bringing it down again.