Factual error: Professional Radio Operators hold the key used for Morse code between their thumb and two fingers - they don't tap it, as was shown. Tapping would produce a harsh voice in Morse code.
Continuity mistake: When the band goes out on deck to play as the ship is sinking all they have on is their uniforms. They are seen this way a few times, in the last shot of the band they all have on overcoats.
Revealing mistake: If you look at the deck planks to the right of Rose and Jack you can see that there is something that runs perpendicular to the rest of the planks. This wasn't on the real Titanic. This is where the set was "cut" so that the front half could be sunk into the tank later on in the film.
Revealing mistake: Throughout the whole movie, you can see the outline of hills in the background, even though the Titanic sank in the middle of the Atlantic.