Matt Damon, as Private Ryan, doesn't have a great deal of screen time compared to the other major roles in the film. He's been captured, after all, and the focus is all on his rescue, and the people risking their lives to rescue him. That means there wasn't a huge amount of scripted dialogue for Damon. He didn't let that stop him making an impression, though.
You may recall a scene in which he tells a story about spying on his brother. It's touching and provides us with a rare insight into his character. It's also completely improvised. None of the dialogue was in the original script, and Damon made it all up on the spot. Spielberg watched it back when making the edit, loved it, and decided to keep it. We guess that's what happens when you include great writers within your cast.