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.
I just saw the movie Crash Dive 2 (which is also about submarines and also involves an enemy Akula-class sub) and it has many recycled scenes from Crimson Tide. To name a few: the scene where the sub does a "snap shot" of two torpedoes, the scene of the Akula being hit, and the scene where a torpedo barely misses the heroes' sub.