Continuity mistake: Back on the boat after the tiger attack, Chef freaks out and yells, "Yes I've had it" and the surfboard behind Chief Phillips' head moves about a foot to the left between shots as he says, "Tiger?" (00:59:40)
Continuity mistake: As Captain Willard smokes his cigarette ,Dennis Hopper says, "I'm curious. I'm very curious. Are you curious?" The captain has the cigarette between his lips pointing upward, but the in the next shot the cigarette is pointing downward and the ash is longer. (02:43:40)
Visible crew/equipment: After Clean is killed there's a shot of Lance sitting on the back of the boat holding a mirror in his right hand. This is not the actor Sam Bottoms but a longer haired thinner stand-in who's not wearing the black wristwatch that Sam wears in every other scene in the movie. (01:54:10)
Continuity mistake: Captain Willard picks up one of the death cards that Lt. Col. Kilgore threw on a dead body and it's the seven of hearts. This is not one of the cards that Kilgore threw down though. As he was naming each one as he threw it and Captain Willard was walking close behind him. (00:28:05)
Continuity mistake: In a face on close-up as he talks to General Corman, Captain Willard's scar starts directly under the middle of his right eye. Later at Clean's funeral the scar is shorter and further to the right no longer directly under his right eye. (00:10:30 - 02:00:45)
Continuity mistake: As Clean says, "I gotta tape from my mom," we see a rope tied around the cleat on the starboard side of the boat. He doesn't go near the cleat as he listens to the tape but the next time we see this spot, right when the attacks starts, the rope is gone. (01:51:00)
Continuity mistake: At the beach party LColonel Kilgore stands up and says, "This is the first of the ninth, Air-Cav, son". If you look closely when he stands up the map that they were just looking at is gone. The colonel was holding it, but he doesn't have it nor does Captain Willard. (00:33:40)
Continuity mistake: As they sit in the boat at Kurtz's compound discussing what they should do, Captain Willard tells Chef about the air strike and hands the radio to him with the antenna standing straight up. In the next shot as he grabs it the antenna is folded over. (02:35:45)
Other mistake: Captain Willard inquires from two soldiers manning a machine gun, "Who's the commanding officer here?" The two can't kill the enemies "out there by the wire" so they ask their friend "Roach" for some help. As "Roach" walks over he turns off a tape recorder playing some Jimmy Hendrix riffs. If you look closely at the buttons of the tape recorder, it was never on in the first place. (01:45:50)
Continuity mistake: As the camera pans along from the rear of the helicopter to the front Chef says to Miss May, "I've got every one of your pictures". Watch very closely as the camera shoots through the two windows on the helicopter's door. The Styrofoam head with the black wig on it that Chef asks the playmate to put on is not up against the left window as it is several shots later inside the helicopter. (01:27:45)
Continuity mistake: Roxanne prepares a pipe for Captain Willard and says, "It was the morphine he took for the wounds he suffered in his heart." The shadow of her left hand falls on her face as she holds the needle over the flame. In the next shot, however, the lighting has changed and the shadow is now nowhere near her face. (02:16:40)
Continuity mistake: On the boat Captain Willard reads newspaper clippings about Colonel Kurtz' and thinks to himself, "38 fucking years old." He is sweating profusely, with sweat dripping off of his face and neck. However, his right hand is perfectly dry when he lifts a flashlight and moments later a cigarette. (01:01:00 - 01:02:50)
Continuity mistake: Colonel Kurtz leaves after he finishes reading the Time articles and there's a close-up of Captain Willard's face and chest. He's soaking wet in the close-up but in the next shot, as he collapses trying to leave the room, his face and chest are dry and there isn't as much dirt on the left side of his chest. (02:52:25)
Continuity mistake: The gentleman sitting next to Roxanne in the tan jacket gets up and says, "Look, this is the truth, an egg, the white left, but the yellow stays". If you look closely the lazy Susan in the center of the table holding the oil and vinegar has been rotated between shots without being touched by anyone. (02:07:20)
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: As the chopper pilot circles the courtyard he says, "We've got wounded down there" and a yellow flair is in the center of the courtyard. In the next shot of the soldiers running around down below we see the center of the courtyard but the yellow flare is not there having been replaced by a Vietnamese hat. (00:43:10)
Visible crew/equipment: Just before the Playboy helicopter leaves the USO show there's a shot of the stage from behind the audience and you can see the shadow of the cameramen moving from left to right on the shirts of the soldiers. (01:10:15)
Continuity mistake: The old gentleman sitting across the table from Hubert de Marais says, "Maybe in the future we can make something with the Vietminh" and the gentleman to the left of Roxanne replies, "Don't you understand". If you look closely the cigar that he just put into the ashtray in the previous shot, along with its smoke, has disappeared. (02:07:00)
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)
Continuity mistake: After the helicopters land on the beach at Vin Drin Dop a soldier yells, "I'm not going, I'm not going" and we see huts that were blown up and burned down earlier in the scene untouched and as good as new. (00:42:50)
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.