From the conning tower we see Cmdr. White walking toward the hovering helicopter that will take him to the carrier, the rescue harness is laying on the deck. In the next scene White has to reach up and grab the harness as it is now dangling in the air.
The ribbons on William Holden's khaki uniform are different than the ribbons he wears on his blues.
On at least three occasions the sailors call the Tigershark submarine a "ship". All submariners know that submarines are referred to as "boats".
While talking with the rescued pilot in the wardroom, LCdr White says he had made three cruises on another boat. During WWII these were called War Patrols, or Patrols for short.
In the beginning when Cmdr. White steps on board his moth balled sub the sign on it says SSN 305. Although it's a fictional movie there really was a SSN 305 which was the USS Skate. The Skate served in and survived WWII. There has never been a USS Tigershark in US history.
In the opening scenes when a down pilot is being rescued, the sub is at full stop. However in the dialogue, the captain is still calling orders to the engine room to slow engines.