Visible crew/equipment: By the cliff, Indy's hat comes rolling towards him. Watch closely and you'll see a long shadow a meter above the ground, parallel to that of the crew member throwing the hat.
Other mistake: In the scene with the tanks, Indy rides up beside one with a rock. Watch the gun closely. The end of the one that he is supposed to shove the rock into is wobbling inconsistantly with the rest of the gun, or the tank for that matter.
Revealing mistake: When the VW gets stuck on the front of the tank and is shot off, you can see that the car has no engine, no body in the car, and no clutch. Also, when the car is blown away you can see a wire used for the stunt fall right in front of the camera.
Continuity mistake: The back of the boat gets chewed up by the prop in the same place twice.
Revealing mistake: When the car is chased by a plane and enters a cave, check the rocks above the entrance and you'll notice that both the entrance and exit are the same location and the final shot has been flipped.
Factual error: When the soldier from Hatay attempts to pass through the booby trap and his head is chopped off there is no blood. Any wound to the head or neck tends to bleed profusely but the head rolls by without a drop.
Revealing mistake: At the end a shadow appears on the cliff under Indy's shoe when he steps to the edge. This would not have happened if a solid but camouflaged bridge were there. (01:47:30)
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.
Other mistake: In the motorbike chase scene there is a lot of flying road dust even in front of the bike in the lead. Probably from the camera vehicle.
Visible crew/equipment: When Indy Sr. is shot, two big stage lights are reflected on his glasses.
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.
Revealing mistake: In the temple, when Indy walks on air, he briefly becomes semi-invisible due to the matte effect.
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.
Other 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: 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.
Revealing mistake: Just after Vogel is thrown out of the blimp, in the shot of him shouting towards it whilst it is taking off, it is apparent that the sky and blimp shown in the background behind him are a projection. (01:13:20)
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: In the wide shot of the Nazi convoy, Colonel Vogel can be seen walking alongside a troop truck. In the next shot, Vogel is no longer there. A few seconds later, the colonel appears on the right side of the screen. (01:23:49)
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.