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)
Visible crew/equipment: After Young Indy locks the two treasure hunters in the reptile train car he pulls a snake out of his shirt, and just as the third young treasure hunter tackles Indy we can see a crewmember's arm and equipment at the left side of the screen, before it cuts to the long shot. (00:07:00)
Revealing mistake: The stuntman who saves Indy's dad from the tank has a perfectly trimmed beard and a thinner face than Sallah's.
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).
Revealing mistake: When Indy and Kazim are fighting on the front of the boat, the front of the boat doesn't move at all when the huge propeller hits the rear of the boat.
Revealing mistake: The candles that adorn the wall in the Grail room do not appear to have a strong enough flame to put out the light they do that shines on the wall. In fact, it looks like there are electrical lamps just behind the candle. (01:49:20)
Factual error: When Indy and Marcus fly to Venice, the route map shows them going from New York to Botwood, Newfoundland, then from Botwood to the Azores to the continent. There were two transcontinental air routes at the time across the North Atlantic: one from New Work to Botwood to London, and one from New York to the Azores to the continent, but none from Botwood to the Azores. (http://people.hofstra.edu/geotrans/eng/ch3en/conc3en/earlyairnetworks.html).
Revealing mistake: When Elsa falls through the crack in the hole, she is swallowed by a big cloud of smoke in the void. However, her silhouette is visible underneath the smoke, revealing she never falls any deeper, and thus giving away how shallow the hole was.
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: 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: 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.
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.
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.