These are the type of changes that really change the story. If you are seeing the movie for the first time right after reading the book, these will pop right out at you.
The British fleet neither seen nor mentioned. In the book, half of the "good guys" in the Atlantic are from the British fleet, who were heading home from an exercise when the story begins. Ryan finds himself on a number of their ships, and a British officer even accompanies him onto the Red October. In simpliying the story, all the British became Americans.
The encounter with the Konovalov was moved up to occur while the Russian sailors were in view.
The October itself is never hit with a live warhead, but the Konovalov, instead of impacting with the October, is hit by one of its own warheads, providing the explosion that they think is the Red October.
The movie does not announce what Ryan's plan is for convincing the sailors that the Red October has been sunk. In the book, the Ethen Allen, a decomissioned ship, is exploded.
In the book, Ryan never boards the Dallas. The messages to the Red October are sent from the HMS Invincible.Minor Changes
The little things, not even worth the red star. There are so many, so I'll just list those that are obvious (to me).