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.
Continuity mistake: When Sallah rides up to the tank to rescue Henry we see Indy toss the whip over the side of the tank, but after the tank falls over the cliff, when Indy climbs back up then walks over to stand between Henry and Sallah the whip is coiled and back in its place, hanging from his belt. Even Indy can't do that. (01:36:00 - 01:37:40)
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: 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: 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.
Factual error: The evil SS officer is addressed as "Herr Oberst" (Colonel) throughout the movie. In the SS, however, this rank was called "Standartenfuehrer". (00:56:40)
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)
Revealing mistake: The biplane sequence is obviously filmed in a studio close up. Despite the fact that they are miles high and swooping around at high speeds, neither Indy's hat, nor his Father's, blow off or even flap in the wind. (01:16:55)
Revealing mistake: When Indy is being chased on the motorboard the guy behind tries to shoot him with a machine gun. If you watch the machine gun carefully it you will see it starts to disintegrate in the guy's hands and the barrel drops off. (00:38:45)
Revealing mistake: When fire sweeps through the catacombs under the church, you can see that the "rats" are completely fake. At the far left of the screen (widescreen) you can see several fake rats just hanging on the wall and not moving at all. In the "special features" section on the DVD, Steven Spielberg admitted that the rats were fake - he couldn't bring himself to roast live ones so he could not make this scene appear realistic.
Factual error: In various scenes, Dr. Elsa Schneider (Alison Doody) is wearing the M43 (Mountain Cap). That cap was introduced in 1943.
Factual error: When Indy is wearing the uniform of a German Artillery Officer, he is wearing the ribbon of the 1939 Iron Cross 2nd class in his buttonhole. The movie takes place in 1938. (01:09:05)
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: 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.
Audio problem: When Indiana is fighting one of the bad guys on top of the rhino's car on the train, the rhino sticks his horn straight between Indy's legs. Indy says, "Holy smokes" but if you look, he actually says something else. (00:07:35)
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.
Visible crew/equipment: After Elsa gives Donavan the false Grail, he turns and you can see the reflection of the camera lens in the cup. (01:51:45)
Continuity mistake: When we first see the knight guarding the Grail, his cloak is tied off with a braided rope, which hangs down on his right side. It stays this way until he says "You have chosen wisely," where the rope has somehow shifted to his left side. The rope goes back to normal when the knight waves goodbye at the end of the movie. (01:49:10 - 01:54:05)
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.