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.
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: 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.
Continuity mistake: When Elsa picks up the Holy Grail and tries to remove it from the Temple, Henry places his tie around the collar of his shirt, but in the next shot, he is now holding on the tie and places it around the collar a second time.
Continuity mistake: When Elsa says "How dare you kiss me!" and starts kissing Indy, her hand is under Indy's ear, but when the camera angle changes to behind her in the next shot, her hand is now pressing his ear.
Visible crew/equipment: At the hotel where Indy knocks on Elsa's bathroom door, she opens it and shuts the music off. The camera pulls backward and its shadow is visible on the left side of the door frame. It's fairly large.
Continuity mistake: During the boat chase in Venice, one of the boats blows up. A small piece of it with a motor comes flying out. In the next shot which is a wide shot, that piece of boat is gone.
Continuity mistake: The tank turret is at a completely different angle when it shoots at Jones when riding his horse.
Continuity mistake: Indy brings the carbon copy to Markus who says "The name of the city is Alexandreta", and a close-up shows the paper with no hands on it; a shot later Marcus' hand is on the paper.
Continuity mistake: The holes produced by the rhino's horn on the roof of the train carriage disappear in the wide shot when young Indy grabs onto the swinging pole.
Continuity mistake: When young Indy fights a baddie, the rhino sticks his horn through the roof and the baddie is pushed to the right. When the angle changes he is falling on the opposite side.
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.