Revealing mistake: When Sallah rides up to the tank to help, just as Indy shouts, "Get Dad!" in the closeup and wideshot of Henry we can see that he's actually lying on protective support, and not lying directly on the moving tank treads. (01:35:40)
Revealing mistake: The stuntman who saves Indy's dad from the tank has a perfectly trimmed beard and a thinner face than Sallah's.
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)
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.
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.
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.
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)
Revealing mistake: In the temple, when Indy walks on air, he briefly becomes semi-invisible due to the matte effect.
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)
Revealing mistake: When Marcus Brody is knocked out in the library, he is dragged away by two of the assailants. You can see the unconscious Brody give his right hand to the assailant.
Revealing mistake: The seagulls on the beach (when Dr. Jones scares them with his umbrella) are blatantly fake. They don't move an inch. This was also confirmed by producer Robert Watts in this making-of video https://www.youtube.com/watch?v=BMFMKsrXuGY (minute 2'30").
Revealing mistake: In the shot of Elsa falling to her death into the misty chasm in the Temple, you can see that it is not Alison Doody but a stunt double.
Revealing mistake: When the tank shoots-up the jeep the interior is revealed to be empty and such a prop.
Revealing mistake: Dr. Jones fires at the jeep from the tank, but the explosion doesn't come out the cannon, nor by the jeep's window next to it; instead the back of the jeep explodes.
Revealing mistake: When Walter Donovan is dying after drinking from the wrong cup, watch the over-the-shoulder shot where we see his hair grow long suddenly - for some reason, this shot was filmed in reverse, as the fire coming from the candles/torches behind Elsa is not burning upwards, but appears to be "sucking" back downwards.
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.