Continuity mistake: When Indy leaves the steering wheel to stop the baddie on his motorboat, he passes by a metal turret twice in less than half a second.
Continuity mistake: During the fight on the tank, Indy punches a German who falls on the wheel and whose hat falls near the turret. When the angle changes the hat is in front of him, but when he falls off the tank the hat is gone.
Continuity mistake: The plane explodes at the exit of the tunnel. A frame later the exit is clean and the small remains are scattered around.
Continuity mistake: When Indy and Elsa encounter the rats, hundreds of them block the way, making Indy lift up his legs. However, a frame later, when Elsa starts to walk, those rats have disappeared.
Continuity mistake: After escaping from the turning blades, Indy's hat swaps from clean to filled with spiderwebs. Though there are 4 seconds in between, he didn't move from the previous spot, nor were there any webs around.
Continuity mistake: While in the train station in Iskenderun, Marcus passes by the same spot and the same people twice (watch the man with the hen on the left).
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: While the propeller chops the boat, Indy grabs Kazim and lifts him up saying, "This is your last chance." But from the opposite angle Kazim is still seated.
Continuity mistake: When young Indy discovers the thieves in the caves, the pot on the right swaps from laying horizontal on the ground to slanted and back to horizontal, depending on the angle.
Continuity mistake: Young Indy grabs the Coronado cross, but from the next angle his hand is away about to grab it, and also its position has changed.
Continuity mistake: When Indy escapes out of his office through the window, there are no shadows around. Half a second later, from a different angle, two long shadows have appeared underneath the window.
Continuity mistake: At the very end of the movie, when they all ride off through a tunnel, you can see all sorts of tassels hanging off each of the horses. When they reach the end of the tunnel and are riding off into the sun, there are no tassels at all apparent.
Continuity mistake: The blood on Indy's left ear disappears at the end of the film.
Continuity mistake: When Indy kisses Elsa in her room in Venice, you can see there is nothing on Elsa's left shoulder, but when the camera angle changes in the next shot, Indy now has his hand on her shoulder.
Continuity mistake: Indy slams a pole against the big X on the floor and produces a hole. When the angle changes the shape of the hole is completely different.
Continuity mistake: While in the zeppelin, the German officer extends his baton and lowers Dr. Jones's newspaper, wrinkling it. From the opposite angle, the newspaper is still raised and the officer repeats the previous movements, though this time he doesn't wrinkle the newspaper.
Continuity mistake: Hitler walks surrounded by a huge crowd ahead holding Nazi pennants. When Indy bumps into Hitler, half a second later, the pennants on the left have disappeared.
Continuity mistake: Dr. Jones hugs his son and walks 20 meters away. When he turns around to address him, he is barely 5 meters away.
Continuity mistake: When a jeep gets stuck to the tank's cannon, the type of jeep, its position and the damage done changes between the first and second shot (watch how the smashed windows fix themselves, for example).
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.