Continuity mistake: When the white robed men ambush the Nazis just after they blow up Salah's car, there's a shot over a Nazi's shoulder as he fires at one of the white robed guys running up a hill. The Nazi's machine gun is held straight out and doesn't move but the dust that flies from the squibs suggest that he fired as he moved the gun right to left which he did not do. The odd thing is that AFTER the squibs go off, the Nazi turns his gun to the left. (01:25:55)
Continuity mistake: When Indiana is drinking champagne with the businessman in the fancy apartment he holds his glass by the stem in one shot and then by the bowl in the next shot. As the conversation continues the glass goes up and down in Indiana's hand like a yo-yo.
Continuity mistake: When Indy and Vogel are fighting atop the tank we see the soles of Vogel's boots and there is a red hexagon dot under each heel, but while Indy is hanging off the tank's side mount gun the soles of Vogel's boots are different.
Continuity mistake: On the tank a soldier jumps from Indy's right to attack him. The following angle shows the soldier coming from the front.
Continuity mistake: When Indy and Elsa are wading in the petroleum looking for the knight's tomb, Elsa unbuttons and opens her jacket and starts to take it off. In the next shot she opens her jacket and takes it off again. (00:34:15)
Continuity mistake: In the beginning where Indy is young, he fights some guys on that zoo train. When he's on the crate with the rhino in it, the background changes in each shot. Once there are trees behind and a split of a second later there are rocks, once there's a cliff and once a plain, and it all changes in splits of seconds as they change shots. (00:07:25)
Continuity mistake: When the Nazi Colonel discovers Henry Jones sen. aboard the zeppelin, Indy asks him for his flight ticket. In the first shot, Indy is standing right beside the colonel (the camera is behind the colonel and Indy comes in from the right), in the subsequent shot, Indy is standing behind the colonel.
Continuity mistake: While in the Germans tank, the right side of Marcus's jacket is bent and wrinkled. A Frame later it looks perfect.
Continuity mistake: Indy and co. leave their horses a way from the steps to the temple, but when we cut to a closer shot they're very close to the entrance.
Continuity mistake: In the Grail temple, when Salah says "Drop your guns please," he is holding a rifle. When Indy instructs his father a few seconds later to get up, Salah is not holding a rifle but has his hands crossed.
Continuity mistake: In the scene in the temple after the two soldiers attempt to get past the first task, the first soldier is seen on his back holding his head beside him. When Indy attempts the first task, both dead soldiers are on their stomachs with no heads in sight.
Continuity mistake: At the airport, someone is reading a newspaper. Instead of the year 1938, the date of the newspaper is 1918 (a guess: I think this mistake happened because the art director wrote the date in a rush, stretching the "3" in a way it seemed a "1" when the people of "atrezzo" saw it).
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.
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.
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.