Continuity mistake: When young Indy says, "Cortés gave away the cross in 1920.", the thief is holding it with both hands. A frame later he holds it with just one hand.
Other mistake: When Indy is atop the tank with his father, he is facing his father. Suddenly, they show a close-up of Indy's face, head-on, with the Nazi leader behind him swinging a shovel towards Indy's head in the background. Indy gasps, and ducks. How did Indy see the shovel being swung at his head if it was completely behind him?"
Factual error: In the 'The Last Crusade', the Germans and their lackeys use MP 38 or MP 40 submachine guns extensively. The MP 38 was still a new weapon in 1938, and not around in such large numbers to arm each and every soldier with it (total of 1 million MP38/40's vs. more than 20 million troopers in WW2).
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.
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.