Continuity mistake: When Indy shoots the 3 Nazis while rescuing his dad, one of them falls and breaks a bench and lands on his right side. When they leave the room, the bench is intact and he's on his left side.
Continuity mistake: When Indy sees Elsa losing her grip on the tilted earth and then falling on the rising ground during the temple quake sequence towards the end of the film, he lets go of his father, but when the camera changes to behind Elsa in the next shot, Indy is back holding his father again.
Continuity mistake: After young Indy escapes from the lion, when he's on the rooftop of the carriage a bad guy aims a gun at him. A shot later his arm is already lowered.
Continuity mistake: On the ship near the beginning, after Indy drops the cross, water comes up and washes it towards the edge of the ship. It goes over a lip on the deck, inches away from falling over the side. However, when Indy goes to grab it, it's not over the lip. It's half on and half off.
Continuity mistake: When Elsa picks up the Holy Grail and tries to remove it from the Temple, Henry places his tie around the collar of his shirt, but in the next shot, he is now holding on the tie and places it around the collar a second time.
Continuity mistake: When Elsa says "How dare you kiss me!" and starts kissing Indy, her hand is under Indy's ear, but when the camera angle changes to behind her in the next shot, her hand is now pressing his ear.
Continuity mistake: During the boat chase in Venice, one of the boats blows up. A small piece of it with a motor comes flying out. In the next shot which is a wide shot, that piece of boat is gone.
Continuity mistake: The tank turret is at a completely different angle when it shoots at Jones when riding his horse.
Continuity mistake: Indy brings the carbon copy to Markus who says "The name of the city is Alexandreta", and a close-up shows the paper with no hands on it; a shot later Marcus' hand is on the paper.
Continuity mistake: The holes produced by the rhino's horn on the roof of the train carriage disappear in the wide shot when young Indy grabs onto the swinging pole.
Continuity mistake: When young Indy fights a baddie, the rhino sticks his horn through the roof and the baddie is pushed to the right. When the angle changes he is falling on the opposite side.
Continuity mistake: When young Indy confronts the lion, the way the whip is rolled changes between shots. The second time a small loop appears.
Continuity mistake: During the rhino scene, the roof of the carriage swaps from dark grey to blue before the rhino's horn goes through the roof.
Continuity mistake: When young Indy falls on the rhino carriage, there's two banners whipping by each of his legs. When the angle changes (to an obvious chroma background) the banner on the right is gone.
Continuity mistake: When young Indy breaks the lion's carriage ceiling and falls inside, a wooden plank falls on his stomach. When the angle changes the plank is gone.
Continuity mistake: When the movie beings, the Boy Scouts at the end of the row dismount the horses, then the angle changes and they dismount the horses again.
Continuity mistake: When the brotherhood member aims his gun at Elsa whilst she is driving in the boat, Elsa ducks downwards to avoid the shot, but in the next angle, she is sitting upright.
Continuity mistake: When Indy breaks through the window of the room where his father is locked up, you can see that it is cloudy and raining outside. However, after Henry says, "Junior, you did it!", in the next shot, there is a very intense and constant brightness (not the instantaneous brightness of lightning) coming through the broken window, as if it were sunny. This brightness disappears two shots later. (00:50:00)
Continuity mistake: When Indy climbs onto the circus train, he begins to be chased by Half Breed and Fedora. However, in the next shot, when Indy enters the reptile car, it is now Rough Rider who is behind Half Breed. (00:06:22)
Continuity mistake: Henry pulls up Indy with both arms, but when the shot changes he is pulling him up with one arm only.
Answer: Henry says, in response to Indy asking if he remembered the details of the trials: "I wrote them down in my diary so that I wouldn't have to remember." So, obviously he did NOT know them by heart. Also, as the other answer says, they didn't want the diary to either be in the Nazis' possession or be burned.