Continuity mistake: When the zeppelin is shown turning around, the biplane that the Joneses are about to escape in is nowhere to be seen. (01:16:10)
Continuity mistake: When Indy takes the mail out of his pocket in the beginning of the film, he keeps the "Grail Diary" in his hand, but puts the letters down twice. Once before he opens the package and once after with the wrapper. (00:23:00)
Continuity mistake: As Indy swings through the window to rescue his father, he gets hit on the head by a vase. His hat is all crumpled and tipped quite far back, then in the very next shot his hat is neat and level.
Continuity mistake: WHen Indy is hanging off the side of the tank, his face becomes completely caked with dirt. After he gets back up on to the tank, his face is relatively clean. (01:34:40)
Factual error: During the motor boat chase through the canals of Venice, Dr. Schneider pilots the vessel between two ships, one identified as the John W. Mackay of London. Although the Mackay was commissioned in 1922, it was stationed at Halifax, Nova Scotia for service in the North Atlantic. It was deployed for the first time to the Mediterranean in 1942 - four years after the purported time of the movie.
Factual error: All through the film, SS-Standartenführer Vogel (Michael Byrne) is seen wearing the Iron Cross in second place beside his SA Military Sports Badge. This is totally wrong. As the Iron Cross is a Military Award, it would be worn over or to the right of any other badge on the pocket. Also presuming this is a 1914-18 Iron Cross, he should then be wearing a ribbon bar over his left pocket displaying his WW1 Honor Cross Ribbon along with long service ribbons, etc. American Law specifically allows the accurate wear of uniforms in movies (see http://uscode.house.gov/uscode-cgi/fastweb.exe?getdoc+uscview+t09t12+443+8++%28%29%20%20AN (subsection f)) and there is no reason to believe (without evidence) that German Law forbids the accurate portrayal of uniforms. (00:53:20)
Continuity mistake: When Indiana is showing Marcus the Grail Diary in his fathers home, he closes the book 4 times in a row without opening it - first 3 times inside the room, and then once on exit.
Other mistake: When Marcus is knocked out at the library, look at his legs when he is dragged away - they are purposely crossed in a way which couldn't have happened naturally for a body being dragged - the actor did it to keep himself comfortable.
Continuity mistake: When Indy and Elsa escape underwater, and surface from the tunnel, Indy's suit should be drenched, but it only has water stains. Then during the speedboat chase, their clothes are completely dry.
Audio problem: In the boat fight scene, Kazim says something like "If you don't let me go, we both die" and Indy shouts "Then we'll die." But if you look at his lips, he's saying something else.
Continuity mistake: In the boat chase, when the bad guys are using machine guns, in one shot we see Indy's boat in the middle of the canal. In the next shot (close-up) they are very close to a large pile of dirt (or wood chips) and in the next (wide) shot they are in the middle of the canal again.
Continuity mistake: When Indy and the protector of the Grail are in the boat about to be destroyed by the rotors, there are panels on the top of the boat that pop up and are destroyed when the boat is moving backwards into the rotors. In the next shot, they are intact, then in the next shot, they are destroyed again. (00:39:20)
Other mistake: When Indiana Jones first enters the castle and speaks with the Nazi officer he is mouthing the Nazi's dialogue as the Nazi is speaking. (00:45:55)
Factual error: Indiana Jones checks his dad's grail diary to find the picture of the stained glass window in the church in Venice. With the book open to that page, the picture is on the left-side page, and under Indy's thumb on the right-side page is a snippet of an article about marriage statistics. The article mentions statistics up to and including the year 1943. Problem is, the movie takes place in 1938. (00:27:00)
Other mistake: In the catacombs, Indy flips over the casket with the skeleton and dives underneath together with Elsa. If you watch closely, the skeleton gets trapped inside the casket, however Indy and Elsa are alone inside the casket. If the skeleton had sunken, there wasn't enough time, nor did Elsa or Indy bump into it when they dove underneath.
Revealing mistake: After Indy jumps onto the tank from the horse there's a shot where he stands up to face the bad guy if you look at the tracks on the sand behind him, they aren't caterpillar tracks but tyre tracks. The reason is that for the close-up shots a fake tank was built on top of a car, a security measure to avoid anyone getting hurt in case they fell off.
Continuity mistake: When Indy and his father are being pursued by German soldiers on bikes, right after they break through the barrier, the barrier guard comes out of his post and waves the pursuing German soldiers to slow down in the background. In the next shot he's no longer there, and the barrier's quite far away, then in the following shot he's back, and Indy's not so far from the barrier again. (01:04:35)
Continuity mistake: In the catacombs underneath the library, when he find the "X" on the wall, he is shown ramming the wall twice to break through. The first time he rams it, however, the wall looks as if he had already had a few goes already. (00:31:55)
Factual error: When Indiana's father uses the plane's gun, he shoots out the tail. This was not actually possible, as the planes had a safety mechanism to stop people doing this. In fact we can see that the Lewis gun he is using is mounted on a Scarff ring, which prevented the trigger of the gun being activated when the barrel was pointed at the aircraft's tail surfaces. (01:17:40)
Deliberate mistake: While Indy is fighting the tank commander, he looks up to see them heading toward a cliff. After the tank goes over, from the time his companions look over the cliff to the time when they walk away and Indy is sitting on the ground, you can see the wind blowing in their faces. But as Indy starts to get up, the wind blows his hat toward him, which it couldn't do as his hat flew off quite a while before, and the winds would have blown it far from the cliff's edge.
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.