The moment of cross-cultural exchange when the defecting Russians bawl their eyes out at E.T. and the other American movies they watch, courtesy of Jonesy.
When Ryan has Dallas send Morse code to Ramius letting him know they know what he wants to do, the look on his face as he snaps the periscope back just screams, "I can't believe it! It's going to work!"
The size of the teddy-bear Jack bought for his daughter, seen at the end of the movie when he's on the flight back to Britain.
It's the same bear John McClane was bringing his kids at the beginning of Die Hard, also directed by John McTiernan.
Also, the fact that he's actually sleeping on the plane, something he was unable to do earlier. It shows just how tired he is after his ordeal, and it's heartwarming because you know he's earned his rest.
Ryan and Ramius' conversation in the end:
Ramius: Do you still like to fish, Ryan? There's a river not unlike this one in Vilnius, where my grandfather taught me to fish. "... and the sea will grant each man new hope, as sleep brings dreams of home." Christopher Columbus.
Ryan: Welcome to the new world, sir.
Ramius and Borodin's conversation about starting new lives in America. Borodin plans on living in Montana, while Ramius hopes to go back to fishing. Becomes a Tear Jerker when Borodin is killed before reaching America.