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: In the catacombs, Indy finds the tomb and blows dust away from the shield, cleaning the sides of the engraved cross, but leaving the rest very dusty. A frame later, the whole cross is spotless.
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 Indy is in the cliff reaching for the grail, his hand swaps between clean, dusty and sweaty, between shots.
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.
Continuity mistake: Henry pulls up Indy with both arms, but when the shot changes he is pulling him up with one arm only.
Visible crew/equipment: After Dr. Jones is healed, Indy tells him to stand up. Stage lights are reflected in Dr. Jones's glasses.
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.
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.