Continuity mistake: When Indy is stepping on the letters, he steps on the "O" and knocks the "P" loose. In the next shot, the "P" is intact.
Continuity mistake: When Indy comes face-to-face with Hitler he hands over Henry's diary with its black elastic around the back cover, but in the closeup as the signature is being signed the elastic is around the front cover, then it's around the back cover again when Indy takes the diary back.
Continuity mistake: In the temple in the Canyon of the Crescent Moon: After Harrison Ford cures Sean Connery by pouring water from the grail over him, Sean stands up with the grail in his hand. In the next shot, however, Alison Doody picks the grail up off the floor. (01:55:50)
Continuity mistake: When Indy and Vogel are fighting atop the tank we see the soles of Vogel's boots and there is a red hexagon dot under each heel, but while Indy is hanging off the tank's side mount gun the soles of Vogel's boots are different.
Other mistake: While Indy and Henry are aboard the zeppelin Indy tosses Vogel out the lounge window, and when Vogel shouts up to the departing airship, in the next exterior shot we see the zeppelin airborne against a scenic view. Later, when Indy realises the zeppelin is turning around it cuts to another exterior shot of the airship, and the scenic view is identical to the earlier shot, only it's been flipped. The images of the zeppelin facing two different directions were composited onto the same scenic background.
Continuity mistake: During the temple quake sequence towards the end of the film, Elsa fails to maintain her grip on the rising earth and falls onto tilted ground, sliding towards the chasm. At this point, Sallah is kneeling on the ground, but in the very next shot, he is now grabbing hold of Henry.
Revealing mistake: When Elsa falls through the crack in the hole, she is swallowed by a big cloud of smoke in the void. However, her silhouette is visible underneath the smoke, revealing she never falls any deeper, and thus giving away how shallow the hole was.
Revealing mistake: When the car is chased by a plane and enters a cave, check the rocks above the entrance and you'll notice that both the entrance and exit are the same location and the final shot has been flipped.
Visible crew/equipment: In the catacombs after Indy passes the picture of the Ark Of The Covenant and approaches the next wall with the lighter, (when the screen goes upwards a bit) the shadow of a boom mic can be seen dropping into view at the top of the lighted wall. (00:31:40)
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.