Continuity mistake: When Donovan turns into a living corpse the badge on his lapel disappears.
Continuity mistake: When Donovan drinks from the grail his left hand holds the base with extended fingers. A frame later they are bent.
Continuity mistake: When Indiana's dad blows up the truck beside them during the chase, there's a big explosion. However, in the next shot showing Indiana and Ernst Vogel, the same explosion behind them is smaller.
Continuity mistake: When Indy and Kazim are about to jump off from the chopped boat, the boat swaps from sinking to floating between frames.
Continuity mistake: The position of the stone that Indy places to block the cannon changes between shots.
Continuity mistake: Right before the castle is on fire, there is a chair very close to the fireplace. When the place starts to burn, father and son take shelter under the fireplace thanks to the fact that the chair has magically moved several meters away.
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.
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.