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: 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: 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 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)
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: 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: 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: After Young Indy runs out of the cave with the Cross of Coronado, when he whistles for his horse and jumps down, note the horse's facial marking and leg marking. When Indy mounts the horse it's a different horse, note its markings, and as he rides toward the circus train the horse has changed again. (00:04:55)
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.
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: 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: Indy falls off the side of the tank and gets his shoulder strap caught on the gun - this should not be possible because his shoulder strap is underneath his jacket, as can be seen when he gets back up onto the tank.
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: At the very end there is a large near circular blood stain on the left sleeve of Sean Connery's shirt. When they ride off, the stain has disappeared completely.
Continuity mistake: When the military caravan with Brody is driving through the canyon towards the Joneses and Sallah, the sun is behind Brody (discernable by the shade). But when Indy is looking down on them - from the front - the sun is also behind him.
Continuity mistake: When Indy takes four horses before battling with the tank, three are black and one is bay (brown with black legs, mane and tail). Sallah is riding a chestnut. When they ride to the Grail Temple, there are two black horses and two chestnuts. Where did the second chestnut come from? (01:28:00 - 01:39:25)
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.
Continuity mistake: In the temple in the Canyon of the Crescent Moon: After Harrison Ford cures Sean Connery by pouring water from the grail over him, Sean stands up with the grail in his hand. In the next shot, however, Alison Doody picks the grail up off the floor. (01:55:50)
Continuity mistake: When Old Dr. Jones is holding Young Dr. Jones while he is reaching for the grail, watch Indy's left hand. In the shot from above, it is by his side, in the shot from below, he is holding the rock. It keeps switching from his side to the rock.
Continuity mistake: When the Nazi is strangling Indy with the chain, Indy looks down into the porthole of the tank and yells, "Dad!" Before he throws the gun down into the tank Indy's hat comes loose, but when the next shot is back on Indy, and he yells, "Dad! Dad, get out!" now his hat is back to normal.
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.