Factual error: Throughout the movie, everytime some type of casualty occurs or an alarm sounds, police type lights go off throughout the ship. Submarines don't have flashing lights or grated decks for that matter throughout the ship. The sound of the alarm is enough, since there are no deaf personnel on board a submarine.
Factual error: When the Captain (Gene Hackman) is getting ready to address the crew, the Chief of the Boat (an enlisted man) is briefly shown barking orders to the crew wearing a combination cover with an officer's golden chin strap. A Chief Petty Officer wears a black chin strap. Contrary to popular belief, it's not illegal to wear an accurate uniform on film, in fact the military offer their help to ensure uniforms are accurate: http://www.defenselink.mil/news/NewsArticle.aspx?ID=516, so that's not the reason.
Factual error: At one point a character sneaks through the bowels of the ship from one end to the other. This would be hard to do since he would have to pass through the lower level reactor compartment. Even if you could do this (there is no such passage on any nuclear sub) it would not be very healthy.
Factual error: At one point, the US sub's starboard (right) side is shown to have an American flag painted on the hull near the bow. The flag's blue field is at top/left when on this side of any vehicle, it should be at top/right. Proper display of the flag requires the blue field always be located 'forward' of the remainder of the flag on all vehicles, be they plane, boat, tank or even space shuttle.