Continuity mistake: At the French plantation dock Lance and Chef get off the boat and it's clear to see that it's not tied to the dock. In subsequent shots sometimes it's tied and sometimes it isn't and in all the close-up shots of Chief Phillips it isn't tied up. (01:57:00)
Continuity mistake: As they pull up next to the Vietnamese sampan, Chief Phillips moves the spotlight on top of the PBR with his left hand so it's pointing at Chef as he takes the papers from the fishermen. Thirty seconds later there's a wide shot of both boats as Chef reads the papers and we see the spotlight back to where it was before facing away from Chef even though Chief Phillips doesn't touch it again. (01:35:05)
Continuity mistake: Chef says to Miss May, "You're beautiful, I just want to " and a white Styrofoam ball appears stuck in her hair between shots. (01:30:25)
Continuity mistake: Captain Willard steals Colonel Kilgore's surfboard from the men sitting in the colonel's helicopter. The helicopter has changed since the colonel and his surfboard flew in on it earlier in this scene. The words "Death From Above" and the two crossed sabers that were painted on the front of the helicopter are not on the front of this one. (00:51:20)
Continuity mistake: In an aerial shot a helicopter pilot says, "I got a vehicle on the bridge " but if you look closely there's no vehicle on the bridge until the next shot. (00:41:40)
Continuity mistake: When the soldiers crash the stage the pink "Playmate of the Year" sign in the back is moving from the wind and then stationary between shots. (01:09:30)
Continuity mistake: Lance wants the surfboard out of his gun turret so Captain Willard pulls it straight up with both hands. In the next shot he's got the surfboard under his arm. (00:54:45)
Continuity mistake: Leaving Hau Phat Clean tells the guys a story about a, "cat in the Delta" and Lance picks up his new water skies then puts them back down again while still holding on to them. In the next shot he's not touching them anymore. (01:12:30)
Continuity mistake: While sitting in the PBR at Kurtz's compound, Chef and Captain Willard discuss what they should do as the Captain puts on a clean shirt. Chef says to him, "He's wacko man. He's worse than crazy, he's evil." The background (box behind captain's head) changes. (02:34:40)
Continuity mistake: When the Medevac helicopter first lands in the Vin Din Dop courtyard there are only soldiers on the left side of it putting the injured man in. In the next shot five soldiers and a Vietnamese woman appear standing on the right of the helicopter who weren't there before. (00:44:05)
Continuity mistake: Approaching Kurtz' compound they pass the natives in their canoes and we see the back of the PBR. Look closely and you can see that the boat's name is painted over. In other shots of the PBR we don't see this mark. (02:28:25)
Continuity mistake: When the woman tries to feed the rice to Captain Willard he absolutely refuses it and the spoon barely touches his lips before she takes it away. The next time we see his face after he spots the colonel there are two small pieces of rice stuck to the right side of his upper lip as he turns to the right after the woman wipes his face. (02:54:00)
Continuity mistake: As they pull up to the French plantation there are several close-up shots of Chief Phillips where you can see Clean's legs lying on the floor board of the boat behind him. As some dead bodies in movies do, they move (six inches to their right) between shots without being touched. (01:56:40)
Continuity mistake: After Chef looks over the papers handed to him by the people on the Vietnamese sampan, he hands them to Chief Phillips, who puts them under a brown flak jacket tied around a metal roof pole on his left. This brown flak jacket appears, then disappears between shots. (01:35:45)
Continuity mistake: As Bill Graham announces playmate of the year Carrie Foster she's still inside the helicopter. Two seconds later in the next shot she's in front of the helicopter standing on the platform that the Green Berets use to carry her into the middle of the stage. (01:07:15)
Continuity mistake: After the tiger attack Chief Phillips asks, "What happened, how many is it?" and Clean is firing the 60-caliber gun behind him at the shoreline. Smoke pours out of the gun as he fires but in the next shot the smoke is not there and the gun has not been fired. (00:59:30)
Continuity mistake: When we first see colonel Kilgore's helicopter there's thick black smoke behind it. In the next shot the smoke is grey and sparse. (00:26:25)
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: When Colonel Kilgore talks to the men and says, "It smelled like victory," a line of sweat and dirt appears on the right side of his chest around his dog tags between shots. (00:49:45)
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)
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.