Factual error: At the start of the movie, the oil rig discovers the first monster. The characters reference the seabed there being at 800 fathoms - which is pretty deep; they'd have to be hundreds of miles or so offshore for the ocean to be that deep. However, the oil rig is just a few miles off a coastline, as the background view shows.
Factual error: Admiral Hadley often barks orders ordering Army forces to act, not just Navy forces. He wouldn't have the authority.