We know more about Norman at peace than we do about Norman at war. At war, we know that he's gentle, but carries a thumb that Mitchell Sanders cut off a VC soldier and gave to him. The only other personal thing he carries is a diary. He keeps telling Kiowa to shut up when Kiowa wants to talk about Ted Lavender's death, but eventually, he's OK with listening. We know, vaguely, that he feels pressure from his father to win medals in Vietnam.
Once he gets home from the war, though, we learn a lot more. Norman is unable to leave the war behind, but he's also unable to talk about it. His father is silent, his ex-girlfriend (now happily married to another man) would just totally not understand, and his hometown seems indifferent in general, but that's not why he can't talk. He just can't. He'll never be able to talk about his harrowing experience in Vietnam, and when he tries to communicate through Tim as a last-ditch effort, Tim fails him. Shortly thereafter, Norman kills himself.