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.
Visible crew/equipment: At the hotel where Indy knocks on Elsa's bathroom door, she opens it and shuts the music off. The camera pulls backward and its shadow is visible on the left side of the door frame. It's fairly large.
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.
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.
Continuity mistake: The tank turret is at a completely different angle when it shoots at Jones when riding his horse.
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").
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.
Other mistake: In the fighter plane scene, their hats don't blow away.
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: 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.
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 Indy and Elsa come up through the manhole their hair is sopping wet and messy. As they run to the boat their hair is magically styled and not sopping wet anymore.
Revealing mistake: When Walter Donovan is dying after drinking from the wrong cup, watch the over-the-shoulder shot where we see his hair grow long suddenly - for some reason, this shot was filmed in reverse, as the fire coming from the candles/torches behind Elsa is not burning upwards, but appears to be "sucking" back downwards.
Revealing mistake: In the shot of Elsa falling to her death into the misty chasm in the Temple, you can see that it is not Alison Doody but a stunt double.
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.