The Russian Crew has been rescued by the American Navy after they bail due to a "radiation leak". Then they believe their captain is fighting the Americans and cheer... on the deck of the American Warship that saved their butts. Not exactly polite. One has to wonder how the American crew reacted.
At one point the Soviet ambassador (played by Joss Ackland) is speaking to NSA Jeffery Pelt. Sweating profusely in a state of agitation, the ambassador tells Pelt that Captain Ramius has suffered a mental breakdown and will try to fire his missiles at the United States. This is a lie, of course, since Ramius is trying to defect and the Soviets want America to help them sink him before he can do that. When called on the fact that he hadn't mentioned this development before, the ambassador lamely blames Moscow for not telling him. Despite his nervousness, the ambassador lies well enough to get the Americans to go along with it . . . except that maybe the ambassador didn't actually lie. If you want someone to tell a lie well, it's helpful if they believe the lie themselves. Given the USSR's reputation for dishonesty, it's not hard to imagine that they decieved the ambassador himself so he would make a more compelling case to the Americans. And that's why the ambassador is sweating so hard. He believes that Ramius really has gone nuts and is about to nuke them, with the most likely target being right where he is sitting.
I've always thought that was what's going on. To make sure the ruse was complete, they just lied to their ambassador (and unseen, probably told at least the ambassadors to the UK and and France and Canada the same lie), and possibly told their naval captains the same story as well. Maybe only a dozen people actually knew Ramius' intentions.
Another iteration - even if they had told the truth to their ambassador, he was so used to his bosses being liars that he didn't believe them, or at least did not know what to believe. If you think about it, everybody in the story except Ramius and those who know him best (his fellow officers and Ryan) is in the position of not knowing what to believe.
Ramius encouraged his men singing the Russian National Anthem - why? Because he knew they would give the sub away. Foreshadowed by Jones' Pavarotti story earlier in the film. Sure enough, Jones hears them singing.
"It was Paganini. Pavarotti is a tenor, Paganini was a composer. If you're going to tell it, tell it right." :)
In the novel, there is a scene where one of the many attack subs, the ES Politovski, undergoes a critical reactor failure that destroys the sub. In a rather obtuse way, this was foreshadowed earlier in the book, during the section describing Ramius' life. It stated that he had been baptized as a Roman Catholic but it could never be revealed, even to him, as 'Marxism-Leninism is a jealous god that allows no others'. The scene involving the Politovski has the zampolitoverriding the chief engineer's request to back down on the dangerously overloaded reactor for repairs. The engineer protested in his mind about 'the Party thinking that physical laws could be overturned by the whim of an apparatchik'. In a round about way, Marxism-Leninism (as a godhood proxy as Ramius described it earlier) was being stealthily paralleled to a theocracy, where the fiat of the priests of the faith (or in this case, the Zampolitni of the Party) trumped everything, even nature itself. An irony there, considering the religious suppression in the Soviet era.
That's not much fridge brilliance. The novel is based on the official version (both in NATO and the Warsaw pact as it was convenient for both sides) of the mutiny led by Valery Sablin in 1976. Except it's very loosely based on this, as in reality, Sablin was a committed communist, was russian, was the ship's political officer, had the captain imprisoned, and was heading to Leningrad to make a radio address to the people of the soviet republics to rise up against the corrupt gerontocracy for a second communist revolution that would restore power to the soviets (more or less the general gist of what survives). He would have hated Ramius.
Connery as Captain Ramius has a bit of Fridge Brilliance to it above and beyond it being, well, Sean Connery. Raimus is Lithuanian - and, therefore, would be presumed to have a fair bit of an accent in Russian, compared to a Great Russian "native" speaker of the language. So, when applying the Translation Convention to English, Connery's Scottish burr reflects that.
Here's something a bit unsettling. Ok, so the Russians tell the US that Ramius is insane, and is getting ready to launch his warheads on the US. Now, let's think for a second, how many people in the Soviet Union were told this? Virtually none. Now, imagine that Moscow got what they wanted, and the US sunk the Red October, killing all on board... and then that information got back to the people of the Soviet Union... To them, the US just destroyed one of their submarines unprovoked, in international waters. The families of the crew (ESPECIALLY the ones with high enough political connections) would be calling for blood. Which, in itself, would spark a war. Not only that, but both Soviet, and NATO ships were within a hairs breath of one another, and during that political fiasco, tensions would likely be tight (something that's even pointed out in the film, three times). Now, could the USSR and NATO reach an agreement to avert a war? More then likely, but the tensions would be so tight it wouldn't take much to spark one. So basically, Ramius faking the reactor problem, getting his crew off in sight of an American Warship (that would likely attempt to stop them from submerging anyway, but wouldn't leave the crew stranded), and telling one of his officers that he intends to scuttle the ship, actually AVERTED a major war. Although the spectacle of a glorious Last Stand was certainly not planned, Captain Tupolev certainly helped cement that one.
Ramius visibly shows great concern for the lives of those of his crew not in on the plot. For instance, he refuses to evacuate the submarine off the Labrador coast, even though that would be a safer option for him, because the waters there are so cold that it is less likely that the men would be rescued alive. However he ends up killing the entire crew of Tupolev's submarine, most of whom were merely mooks doing their duty and were just as innocent and undeserving of death as his own crewmen were.
Oh yes, and he really might have started a nuclear war. It is stressed more in the book than in the film just how likely it was that one of the numerous tense confrontations between NATO and Warsaw Pact forces that resulted from the pursuit of Red October would spiral out of control.
Rationally, the Americans ought to have hunted down and destroyed Red October even if they did have a hunch that Ramius was trying to defect. Even the slightest chance that he might be planning an independent launch of his missiles - which would have killed millions - should have been too grave a risk to run.