Continuity mistake: When Elsa's hand slips from its glove whilst Indy is holding on to her, Elsa falls and she is holding the glove in her right hand, but when she falls into the chasm, the glove has gone.
Continuity mistake: While Henry and Marcus are being held inside the tank, Indiana rides in circles and baits the tank to ram into one of the cars. In a following shot when we see Vogel walking toward the window inside the tank it's a flipped shot, note everything is backwards. Then two shots later, when Vogel shouts "quick" in German it's another flipped shot, and about five shots after that again flipped. (01:29:30)
Continuity mistake: When Elsa is looking out over the Canyon of the Crescent Moon, the sun is low - it is close to sunset. However when Indy is riding through the very narrow canyon, the sunlight reaches the bottom. It is not the same time of day (or later the same day).
Continuity mistake: When Senior is rescued from the tank with a whip around his ankle, the whip dissappears a split second later, when he is being mounted on the horse.
Continuity mistake: When Indy is rescuing his father from Brunwald castle, he uses his bullwhip to smash through the window of the room his father is kept in...but in the time it took to jump into the room and get smashed over the head by his father, the bullwhip he used would have lost lots of its momentum and it wouldn't be waiting for him to grab it and use it later on the movie. (00:47:40)
Continuity mistake: When they are first looking over the cliff at the tank, Sean Connery's beard is quite full, and in a shot moments later, still looking off the cliff, it has been trimmed. This is due to the fact that this scene was filmed after principal photography had ended.
Continuity mistake: During the big chase scene leading up to the three challenges, Indy uses his gun and begins to takes the high road on his horse. Harrison Ford couldn't seem to find the holster for his gun, so you can see his holding the gun by his side until the scene changes. (01:31:15)
Continuity mistake: The back of the boat gets chewed up by the prop in the same place twice.
Continuity mistake: After Indy exits through the sewer with his hair all wet, he escapes Kazim's men. When he turns around a block his hair is nicely combed and loose, instead of sticky. Not too plausible when you're running for your life and your head is soaked in petrol.
Continuity mistake: The distance between Indy's boat and the big red and white ship keeps increasing and decreasing randomly.
Continuity mistake: In the zeppelin, after Indy says "no ticket", everyone searches for their tickets and waves it at Indy. When the angle changes, a frame later, the people are searching for their tickets, no one is showing one.
Continuity mistake: The crater Indy's car falls into becomes bigger and way deeper between the first and second shot.
Continuity mistake: When the driver of the tank is shot and he changes the direction of where the tank is going, Indiana gets up, with his strap still stuck on the nozzle. However, in the next shot, he's become clear of being stuck. There's no way he could have gotten his strap undone so quickly.
Continuity mistake: Near the end, in the "Leap of Faith" scene, Jones steps off the edge of the abyss and discovers an ingeniously-camouflaged stone bridge. The camera pans to show the camouflage effect from another angle, and it's obvious that the stone bridge is only about one foot thick from one end to the other. Relieved, Jones scampers across to the other side, and the camera cuts again to show that the bridge is now a full three feet thick.
Continuity mistake: Before the propeller chops the boat, Elsa leaves and looks at Indy. Right then, the camera is reflected on her boat's windshield.
Continuity mistake: During the fight on the Coronado's deck, Panama Hat is climbing the ladder with his back to the camera. In the next shot, when he says, 'Grab him, he's getting away!', he faces the camera. (00:13:21)
Continuity mistake: Inside the Grail's Temple, all the characters have the decapitated bodies of two Hatay soldiers and the two lion statues in front of them. However, when Indy's team is discovered and the shot changes, it can be seen from the position of the bodies and the statues in relation to the characters that they have all changed places, with the bodies now being to one side of the characters (to the left of Donovan and to the right of Indy). (01:41:55)
Continuity mistake: After the Brotherhood of the Cruciform Sword members begin shooting, in the wide shot of the canyon gunfight (with Indy, Henry and Sallah in the right corner of the screen), Vogel is standing behind the tank, shooting at the Brotherhood with his Luger. In the next shot, he's suddenly running with two soldiers. (01:25:47)
Continuity mistake: At the beginning of the movie, in the shot after Roscoe says "He's got our thing!", the lamp behind him and the lamp in front of Fedora disappear. Also, just after Roscoe says that line, immediately his voice is heard saying "Get him!", but his lips don't move. (00:04:44)
Continuity mistake: The remaining Nazi biker passes among his fallen compatriots following Indy. The latter looks over his shoulder at the biker. In the next shot, the background behind the Nazi is totally different (the bodies, the motorbikes and the hut have disappeared). (01:04:53)
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.