Goofs

The deck is not grated (like on a BBQ pit) and there is no "crawlspace" that you can just go down into and sneak around the sub.

Submarines don't have rotating yellow lights inside (like the kind you'd find on construction machinery or something) which flash and are located throughout the movie.

When the woman is telling the president about the two crew member remaining aboard, she says James Piersall is an electrician's mate qualified in weapon's control. Sailors have specialized ratings and therefore someone with a weapon rating would have that qualification and an electrician's mate would work in the engine room (they make reference to a reactor room in the movie) where the reactor that supplies power is controlled and maintained.

When the Montana launches the missile to some empty ocean area, the missile tube doors shown are not in the configuration they should be.

When the Florida shoots torpedoes at the Montana, the Captain orders right full rudder and 20 degree down bubble and ahead full. He may have done so out of arrogance or cockiness, but in the event of that actually happening the bell (speed) ordered would be all ahead flank. Same goes for the Florida right afterwards when the captain orders evasive action all ahead full.

The terrorist Chang "electronics expert" while fixing the electrical box grabs the hot end of the soldering iron. She is good though as she doesn't even scream out in pain.