Continuity mistake: Harrison Ford lights up his cigarette and Captain Willard says, "I'm not presently disposed to discuss those operations sir." Jerry is sitting with his back almost up against the wall. In the next shot as he takes a puff from his cigarette he's moved and is now leaning forward. (00:10:45)
Continuity mistake: As the VC first hear the helicopters coming an old woman in a pink top with a baby runs across the schoolyard. In the next shot she disappears and most of the school children have changed positions. (00:39:05)
Continuity mistake: The paint on Captain Willard's face changes many times throughout the movie. Two very noticeable examples are right after he kills the colonel, where the paint is almost entirely off his face, and several shots later as he leaves the temple heading back to the boat where his entire face is painted a much darker green then before. (03:09:20 - 03:11:50)
Continuity mistake: As they get close to the French plantation dock we can see inside the building and it's dark. Several shots later as Captain Willard is walking around inside the building it's lit up from the inside. (01:56:00)
Continuity mistake: As Chef pulls the Vietnamese woman out from where she's hiding you can see the shadow of Lance's gun barrel as he stands there holding it on her. In the next shot Lance gets up from a seated position in front of the boat. (01:36:20)
Continuity mistake: In the first few scenes with the air cavalry commander, his black hat has a lieutenant colonel rank and a small crossed-sword emblem on it. Later, as his helicopter lands on the beach, his hat has no rank and is emblazoned with a very large crossed-sword Cav emblem. (00:27:26)
Continuity mistake: The Vietnamese woman on the sampan has her hair in two ponytails when Chef shoves her. It's in disarray when she falls then in one ponytail as she is shot by Clean. (01:36:25)
Continuity mistake: The Frenchmen with the accordion asks Captain Willard, "Do you know anything about Dien Bien Phu?". If you look closely at the doors in the other room, most of the lower shutters are closed. The next time we see them all as the Frenchman playing the accordion stands behind the captain and says, "Dien Bien Phu, okay" all the shutters are open even though no one goes near them. (02:08:30 - 02:10:00)
Continuity mistake: When Chief Phillips is speared we first see Lance at the helm of the boat and it's going forward. In the next shot after Chief Phillips is speared the boat is going around in circles. (02:22:30)
Continuity mistake: Lieutenant Carlsen says, "Get that light off me" as the PBR approaches the shore. Lance, sitting at the front machine gun, starts to move the spotlight before the lieutenant says anything then in the next shot the spotlight is already facing him. (01:41:40)
Continuity mistake: Hubert has a discussion with the Frenchmen sitting to the right of Roxanne who gets up and leaves. As he gets up his hands are on table but in the next shot they're at his sides. (02:11:05)
Continuity mistake: Chef and Clean pull Lance into the boat after they ditch Col. Kilgore but in the next shot as Lance climbs in the boat Clean has disappeared. (00:51:15)
Continuity mistake: Captain Willard sits in the sunshine near the back of the boat reading his file on Colonel Kurtz but in the wide shots of the boat as Lance is water skiing he's not there. (01:15:20)
Continuity mistake: After Harrison Ford lights up his cigarette and walks towards the General, we see Jerry sitting behind him putting the piece of paper he was reading into his briefcase. In the next shot Jerry is holding the piece of paper in his left hand again. (00:10:40)
Continuity mistake: As Captain Willard sits in the boat reading and thinking to himself about Colonel Kurtz he unwraps and eats a Hershey bar. He crumples the wrapper with both hands and throws it out of the boat, leaving nothing in his hands. In the next shot he's eating the Hershey bar again and the wrapper is back. (01:15:45)
Continuity mistake: After Willard kills Kurtz, and Willard is leafing through some papers that belonged to Kurtz, he opens the memoirs halfway through the stack. As he closes the stack, the visible page changes to a different page and different placement. (02:22:00)
Continuity mistake: Colonel Kilgore talks about getting the boat up the river during the beach party and says, "Hell, a six foot peak," and a cigarette appears in his mouth between shots. (00:33:50)
Continuity mistake: When they leave there are two fires on the Do Lung Bridge as Clean looks on manning the rear machine gun. In the next shot there is only one fire on the bridge. (01:48:45)
Continuity mistake: After he hears the Frenchmen shouting at them Chef turns around and looks at Chief Phillips as he puts his gun down on the side of the boat. It in the next shot he's looking in the opposite direction. (01:57:30)
Continuity mistake: The next day as the boat leaves the USO show at Hau Phat to continue up the river we see the battered stage from the back of the boat. The boat passes the stage by about 100 feet but in the next shot its back right next to the stage. (01:11:25)
Answer: It's not any specific song; it's just the kind of generic piano music you'd hear at dinner at a high-class restaurant of the era. Think of scenes in movies, films, etc. set or made in the '50s and '60s, where the characters go to a nice dinner and there's someone at the piano playing unobtrusive music to accompany the food/conversation.