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.
Audio problem: Just after young Indy escapes the bandits on the train, he finds the magic cart and says "Magic" to himself, but a close look will show his mouth doesn't move with the line.
Continuity mistake: After Indy and Elsa exit the catacombs through a sewer and start to run away, a woman in pink turns her head to watch them leave and then turns around to see Kazim's men run after the couple. A frame later, from a new angle, the woman's head is turned 45° facing downwards.
Continuity mistake: Sallah rescues Dr. Jones from the tank and stops by the cliff. On the next wide angle he's nowhere around.
Continuity mistake: Indy stops his sidecar at a crossroad, meters ahead of a house. When the angle changes he is now several meters behind.
Continuity mistake: During the crashes by the border control the road keeps swapping from sunny to shadowy back and forth in less than 3 seconds.
Continuity mistake: When Marcus watches the carbon copy of the shield, his ice-bag is on the table, but when the angle changes he has it next to his head and seconds later he puts it on the table. Also, the ashtray moves backwards between shots.
Continuity mistake: On the tank, Indy is punched and the gun falls inside. His hat also falls off, but a shot later it's back on, perfectly tucked in.
Audio problem: When the temple floor cracks, Indy holds Elsa from falling into the void and she gives a very loud cry, but her mouth is closed.
Continuity mistake: After Dr. Jones fires at the jeep two soldiers fly towards the tank, but when the angle changes they're falling away from the tank.
Continuity mistake: Inside the temple, when Indy is at the cliff, in the close-ups the rocks next to him are round shaped, but in the wide angle they are straight and stripped.
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.