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: 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.
Revealing mistake: When Walter Donovan is dying after drinking from the wrong cup, watch the over-the-shoulder shot where we see his hair grow long suddenly - for some reason, this shot was filmed in reverse, as the fire coming from the candles/torches behind Elsa is not burning upwards, but appears to be "sucking" back downwards.
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.
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.