Revealing mistake: At the beginning of the film, Bond is on a training exercise to free a trapped hostage. Since it is only a training exercise, everyone involved is using blanks in their weapons. Why is it that when Bond fires an automatic rifle, we hear multiple bullets ricocheting? (00:03:25)
Revealing mistake: After the opening scene, M is viewing it all on tape. But the tape shows the scene in the same way as the audience saw it - the line of sight is changing, zooming in etc., much more than hidden surveillance cameras could accomplish. (00:04:20)
Revealing mistake: There is an underwater scene where Bond is swimming in a sunken ship, trying to escape a shark. At one point, he swims into a cabin,and the sharks head comes through the door. The camera then goes to a close up of Bond's hand trying to latch the door, while the sharks head is pushing on it. The film is then reversed several times to make it look like Bond is trying to push the door shut while the shark is pushing it open. (00:51:40)
Revealing mistake: During the credits, Bond grabs a rope and jumps inside a house. His stunt is way younger than him and very noticeable.
Revealing mistake: When the movie begins Bond climbs up a telephone pole. From behind he is obviously not Sean Connery: not only his build is different, but his hair is a weird squared wig, maybe some sort of security helmet.
Revealing mistake: Small black ramps are against the kerb for the stunt rider to use to mount the pavement during the motorbike chase.
Revealing mistake: When Domino and Bond escape horseback riding, a man fires a rifle, but no smoke comes out despite the bang being heard. (01:47:07)
Revealing mistake: When Bond rescues Domino from being sold, and she hops on the horse, Kim Basinger is replaced by an obvious stunt.
Revealing mistake: During the shoot-out, all the baddies' heads and hands are bare, except for the one that gets burnt alive, who wears very obvious fireproof hood and gloves.
Revealing mistake: After Nigel and Bond meet and stop walking, there's a group of bystanders behind staring at the shooting of the movie.
Revealing mistake: Before Bond throws the urine specimen in Lippe's face during their fight at the health farm, look at the beaker and you can see a tablet dissolving in the water.
Revealing mistake: When Fatima is blown up by the explosive charge, there is a wide shot of a dummy standing in her place.
Revealing mistake: When Jack's car crashes, the actor with a military hat is replaced with a stunt driver wearing a silver helmet.
Revealing mistake: After Bond throws urine at the thug, and he collapses with all the glasses stuck to his back, note the protruding pad underneath his shirt where everything's been attached to.
Revealing mistake: In the shot of the US submarine from below, the waves above move too fast and it is clear that they are little ripples on the water above a small model boat.
Chosen answer: Neither is a professional dancer. Having studied dancing some thirty years before does not mean someone has retained the same level of skills. As you mentioned, Connery was in his 50s by that time.
raywest