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.
Revealing mistake: When Marcus Brody is knocked out in the library, he is dragged away by two of the assailants. You can see the unconscious Brody give his right hand to the assailant.
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.
Other mistake: During the airplane chase Indy is seen firing his revolver at the German planes, after he crash-lands he pulls his revolver and breaks open the action to check if he's got any ammo left. He doesn't, but the issue is that a revolver doesn't eject spent shell casings, and there aren't any empty casings in his revolver, despite him firing his gun earlier and having no chance to empty the casings since.
Other mistake: After getting on aboard the dirigible, Henry is holding a German newspaper when addressing Indy. But he'd have a hard time reading it as it's upside down.
Continuity mistake: Henry pulls up Indy with both arms, but when the shot changes he is pulling him up with one arm only.
Continuity mistake: When Indy is in the cliff reaching for the grail, his hand swaps between clean, dusty and sweaty, between shots.
Continuity mistake: Henry tells Indy "let it go" and Indy takes his left arm to where his father is. Half a second later his arm is away and he repeats the previous movement.
Continuity mistake: When the grail falls down the cliff and Elsa and Indy try to grab it, the ground it lies on is either yellow or brown. This changes back and forth and has nothing to do with lighting.
Visible crew/equipment: At the beach, right before Dr. Jones scares the birds there's a zoom-in close-up of his face and a crew member approaching is reflected on his glasses.
Revealing mistake: The seagulls on the beach (when Dr. Jones scares them with his umbrella) are blatantly fake. They don't move an inch. This was also confirmed by producer Robert Watts in this making-of video https://www.youtube.com/watch?v=BMFMKsrXuGY (minute 2'30").
Other mistake: In the fighter plane scene, their hats don't blow away.
Continuity mistake: While being chased by the plane, Dr. Jones faces backwards and cries "This is intolerable!", then faces forward and ducks. From the immediate wide angle he is still facing backwards.
Continuity mistake: During the escape from the Nazis the Jones boys pass through a boathouse and Indy sets a small boat loose with the engine running to distract them. When he releases the boat the motor (and boat) is turning to the starboard side (which would turn the boat straight back into the bank), but when the Nazis arrive, the boat is running perfectly straight down the river.
Continuity mistake: When Indy and his father take the car from the old man, you can see that viewed from behind Henry is the driver and Indy is next to him. In the next shot, they have changed positions and Indy is now driving the car.
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.