Continuity mistake: While Indy and his father are talking in the zeppelin, the amount of flowers in the vase keeps changing between shots.
Continuity mistake: When the tank is about to go over the cliff you can see that there is no one sticking out of it, but in the next shot, Vogel can be seen sticking out and yelling.
Continuity mistake: While the German tank travels to the temple, it passes a German soldier. Inmediately after, a very wide angle shows the soldier has dissapeared.
Continuity mistake: Jones Sr. falls on the tank's chain and passes by the turret and by a round rusty thing. When the angle changes, a second before Indy saves him with the whip, you see Sr. has moved half a meter back instead of forward, and is passing by the same elements as before.
Continuity mistake: When Marcus is taken to the temple by the Germans, the knot on his tie differs between a far shot and a close one.
Continuity mistake: When the Zeppeliner turns back towards Berlin, it is over snow covered mountains. But when the Joneses escape in the biplane there is no sign of snow anywhere.
Factual error: When Hitler signs the diary, he writes in a modern way of writing. But in the late 30s there was a different style of writing called "Suetterlin". The letters looked quite different from nowadays. (01:10:25)
Continuity mistake: Indy throws the German officer out of the zeppelin window and Senior stands up to watch it all better. Then he sits down. Angle changes to a wider view and Senior is back on his feet, head almost out of the window, repeating the previous movements.
Continuity mistake: Indy and a soldier fall down on the tank, right where some bags are. From the next angle they are half a meter before the place where the bags are.
Revealing mistake: Just as Elsa leaps from one side of the precipice to the other as the Temple collapses, a rock rolls from right to left and hits her right in the head before bouncing off. (01:56:35)
Revealing mistake: A bomb falls in front of Indy's car, creating a crater where the car falls into. Dr. Jones Sr. doesn't move forward due to the momentum of the crash, revealing he's been replaced by a dummy.
Continuity mistake: When Indy is looking up at the front of the Grail temple it is bathed in direct sunlight. Indy dismounts his horse in the shadow. However, in the shot just before dismounting, the cliffs behind Indy are also bathed in direct sunlight. That is impossible.
Continuity mistake: When Young Indy is hoisted from the lion's carriage on the train, one of the baddies is holding a gun at him, in the next shot, the gun is down by his waist.
Continuity mistake: When Indy gives his father water from the grail to drink, the grail glows from inside, and the gold inside it is bright and even. But when Indy pours water over the wound, the grail is dark inside and there are dark patches in the gold. Obviously, they had to take the light that made the grail glow out so it wouldn't be seen when Indy pours the water over his father.
Revealing mistake: When Kazim dies, his shirt is all bloody, but there are no holes in it. The bloodstains looks painted on. (01:26:30)
Continuity mistake: The 109s that are animated are shaped like the original German fuel injected ones. The shots where it's a real plane show the modified one with a RR Merlin engine that gives it a weird square nose.
Continuity mistake: As Sallah pulls Dr. Jones senior off the tread of the tank, Indy lets go of his whip and turns towards the German colonel and they continue fighting face-to-face. You can see that Indy punches the colonel so that he loses his balance and stumbles backwards. A split second later, the colonel is standing behind Jones with his arm around Indy's neck. They did not have nearly enough time to so radically change positions. (01:32:15)
Other mistake: When Indy and Kazim are fighting on the the boat as it is being chewed up by the giant propeller, the propeller is more than half way out of the water; if the ship was in dry-dock the smaller boats wouldn't be able to go up to the larger ship, on the other hand if it had run aground it would not be running its engines.
Continuity mistake: The lion approaches Indy from the rear of the circus car and walks towards the center. Next shot, the lion approaches from the back of the car again.
Continuity mistake: The Joneses take off in the Zeppeliner from Berlin. Shortly after takeoff they are over tall snowy mountains but there are no tall snowy mountains that close to Berlin.
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.