Continuity mistake: Before they make the second fighter pilot crash, Henry hands his bag to Indy. In that shot, the handle of Henry's umbrella is closer to Indy's left shoulder. But, in the next shot from in front of Indy, the handle is closer to his right shoulder. Then, when Henry pulls the umbrella out, the handle is back near Indy's left shoulder.
Continuity mistake: In the cave, young Indy is giving instructions, his hair is down both sides of his face. When he removes the snake saying "It's only a snake", his hair style is changed.
Continuity mistake: The gap between Indy and the chasing motorbikes changes considerably between shots, close to him than far apart the close again and at the checkpoint out of sight.
Continuity mistake: The dirt road the solo motorbike rider takes heading towards Indy is completely different to the road Indy uses just after he knocks the solo rider off.
Continuity mistake: When Indy is on the boat and says "Too small for two of us!", it pans down and you can see the cross halfway out of Indy's pouch. Both of the arms and head are on the outside. Only the bottom half is in his pouch. The bad guy then proceeds to open the pouch, revealing only the top before taking it out.
Other mistake: After getting on aboard the dirigible, Henry is holding a German newspaper when addressing Indy. But he'd have a hard time reading it as it's upside down.
Continuity mistake: When they first meet, Walter Donovan hands Indiana the glass of champagne almost filled to the brim. The next shot shows Indy's glass with much less champagne in it.
Continuity mistake: Indiana has his satchel on him at the beginning of the 3 challenges and up to the point when he is entering the resting place of the grail. As soon as he enters the cavern where the Knight is, and he stands up, he no longer has the satchel.
Continuity mistake: Young Indy escapes from the four diggers, and you can see them exit the cave after him as he pauses. He moves down to and jumps onto the horse (missing the first time). He gets on the horse and you see the diggers, starting running from the exact same place that they were. It's like they paused their pursuit while the camera was not on them. If they kept running, they would've easily caught up with him before he jumped down.
Continuity mistake: When the boat scene starts, Indy passes by an iron tower. Then he stands to fight the baddie and the boat passes by the tower again.
Continuity mistake: Henry pulls up Indy with both arms, but when the shot changes he is pulling him up with one arm only.
Continuity mistake: When Indy is in the cliff reaching for the grail, his hand swaps between clean, dusty and sweaty, between shots.
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 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: When Henry surprises Marcus in the tank, there is nothing hanging out of the pocket on Marcus' suit. In the next shot, there is now a pendant hanging out of the pocket, but when Marcus shakes Henry's hand in the next shot, the pendant has disappeared.
Continuity mistake: When Indy shoots the 3 Nazis while rescuing his dad, one of them falls and breaks a bench and lands on his right side. When they leave the room, the bench is intact and he's on his left side.
Factual error: When the Nazis are meeting with the Sultan of Hatay he remarks he wants their Rolls Royce Phantom II, he quips about the size of the engine being 4.5L engine with 30HP. In reality, the Phantom II had a 7.7L I-6 engine with 40-50HP. Also, the vehicle shown in the movie is a Rolls-Royce 20/25HP Barker Saloon, not a Phantom II. It also makes little sense that the Nazis would be driving a Rolls Royce since it is a British car. In the novel, the car is said to be a Daimler-Benz, which matches what the Nazis actually used in real life.
Continuity mistake: When Indy sees Elsa losing her grip on the tilted earth and then falling on the rising ground during the temple quake sequence towards the end of the film, he lets go of his father, but when the camera changes to behind Elsa in the next shot, Indy is back holding his father again.
Continuity mistake: After young Indy escapes from the lion, when he's on the rooftop of the carriage a bad guy aims a gun at him. A shot later his arm is already lowered.
Continuity mistake: On the ship near the beginning, after Indy drops the cross, water comes up and washes it towards the edge of the ship. It goes over a lip on the deck, inches away from falling over the side. However, when Indy goes to grab it, it's not over the lip. It's half on and half off.
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.