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.
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)
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.
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: During the boat chase/fight in Venice, the rear of Indy's boat gets shot up and starts smoking and spouting flames and the engine starts chopping. He leaps onto the pursuing boat, fights until that boat is destroyed in the rotors, then leaps with the Grail protector back onto his original boat with Elsa. The boat is now running perfectly. (00:38:50 - 00:39:55)
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.
Factual error: German Fighters shown in the film are in fact Pilatus P-2 (Swiss-made aircraft built after World War II). The appropriate fighter of the period would be the Messerschmitt 109. (01:17:00)
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)
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: 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)
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)
Continuity mistake: The sky colour/condition changes between the shots of Indy and his father in line to get on board the Zeppelin and the scene just before. In one, it is cloudy, and in the other it is a clear blue.
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: When Indy is being chased by the tank there is a type 82 Kübelwagen - they weren't produced until 1940 and the movie is set in 1938.
Continuity mistake: When Indy comes face-to-face with Hitler he hands over Henry's diary with its black elastic around the back cover, but in the closeup as the signature is being signed the elastic is around the front cover, then it's around the back cover again when Indy takes the diary back.
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: When young Indy (River Phoenix) is advised by the fedora-topped henchman at the end of the chase, the cut he received while using the bullwhip in a previous scene changes angles. This is a flipped shot, note Indy's backwards shirt. (00:08:25 - 00:11:45)
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.