Continuity mistake: Captain Willard reads on uninterrupted from "Commitment and Counterinsurgency" by Colonel Kurtz who says in his report, "if they were committed this war can be won with a fourth of our present force". The weather changes from not raining to raining, back to not raining again between shots. (01:15:40)
Continuity mistake: As Captain Willard holds Colonel Kurtz' paper titled, "Commitment and Counterinsurgency" in front of the camera it changes color from yellow in the first shot to white in the second. (01:15:40)
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: When Lazaro's boat initially approaches, the guy on the front has a flare in both hands that he's just about to ignite. When they cut to the next shot he doesn't have anything in his hands. (01:16:15)
Continuity mistake: Two other patrol boats pass them on the river going the other way and Chief Phillips yells over the megaphone "is that you Lazaro?". The guy whose pants fall down has a cigarette in his mouth, and his pants are down further in the front as the boats approach. In the next shot after they pass by, he has no cigarette in his mouth. (01:16:20)
Continuity mistake: The guy on the second patrol boat that throws the flare onto their roof starts out wearing a dark green beret, but in the next shot, when we see him up close, he's not wearing any hat at all. (01:16:30)
Continuity mistake: A guy from Lazaro's boat throws a smoke canister that catches fire on the roof and Chef tries to put it out with a fire extinguisher. When we first see him trying to put the fire out his shirt is dry then the next time we see him his shirt is full of sweat. (01:16:40)
Continuity mistake: As the smoke canister catches fire on the roof and Chef tries to put it out with the fire extinguisher Captain Willard goes from having his back towards Chief Phillips to facing him between shots. (01:16:40)
Continuity mistake: As Chef tries to put the fire out with the fire extinguisher Captain Willard is hanging over the side of the boat with Clean as they try to escape the flames. In the next shot and as chief Phillips walks around him he's behind the helm of the boat near the Playboy picture that Chef taped up. (01:16:50)
Continuity mistake: Captain Willard reads Kurtz' file and narrates, "Late Summer, Autumn 1968" as Clean drums in front of him. At first Clean has both sticks on the helmet than in the next shot the right stick is on the helmet and the left is hitting the boat. (01:17:20)
Continuity mistake: Captain Willard says, "the army tried one last time to bring him back into the fold" and we see a close-up of General Corman again sitting at lunch with the captain from earlier in the movie. This was not shot at the same time because the general's shirt is different. Look closely at his name, the collar covers everything except the first two letters whereas earlier in the movie, for the entire scene his name was showing and could be read. (01:18:10)
Continuity mistake: On the boat while Captain Willard reads Kurtz' file Clean starts making noise by tapping his drumsticks. This annoys the captain and as he gets up to confront him we see, at the right of the screen, the helmet that Clean is drumming on. In the next shot the helmet and its strap have been moved around the metal pole that holds up the roof. (01:19:05)
Continuity mistake: Captain Willard says to Chief Phillips, "how long has that kid been on his boat?" and a long green piece of fabric is dangling off the roof of the boat behind the Chief. In the next shot of the captain, it's gone. (01:19:20)
Continuity mistake: On the boat Chief Phillips says to Captain Willard, "my orders say I'm not supposed to know where I'm taking this boat, so I don't". If you look at the binoculars next to Chief Phillips on the right of the screen they rotate 90 degrees between shots. (01:19:40)
Continuity mistake: The large bell shaped radar antenna is missing in several of the long shots of the boat as they go up the Nung River. It was there when the roof caught fire and we see it back again when they meet up with the Playboy bunnies (redux version). (01:20:30)
Continuity mistake: When the boat stops at the Playboy bunnies' camp it's raining. It's a partly cloudy day and if you look closely it's only raining in the foreground and not in the background. (01:22:30)
Continuity mistake: As they stop at the Playboy Bunnies' camp, two barefoot soldiers run out to meet them. When they first come out there's a red and white oil drum standing in front of their tent. In the next shot it's overturned but later when they walk back into the tent it's upright again. (01:22:55)
Continuity mistake: At the Playboy bunnies camp Captain Willard jumps off the boat with his rifle in his left hand. In the next shot as he turns around it's in his right. (01:23:30)
Continuity mistake: After speaking to Bill Graham in the tent Captain Willard comes back and tells the guys, "I negotiated two barrels of fuel for a couple of hours with the bunnies". As he walks back to the boat the hills in the distance can't be seen because of the rain but in the next shot when Chief Phillips comes over to talk to him the hills can be seen. (01:26:25)
Continuity mistake: At the Playboy Bunnies camps Lance picks up the second barrel of diesel fuel with both hands as he takes it off the boat. In the next shot both his arms are wrapped around it. (01:27:20)
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.