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: 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 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)
Continuity mistake: When Indiana is drinking champagne with the businessman in the fancy apartment he holds his glass by the stem in one shot and then by the bowl in the next shot. As the conversation continues the glass goes up and down in Indiana's hand like a yo-yo.
Continuity mistake: When Indy and Vogel are fighting atop the tank we see the soles of Vogel's boots and there is a red hexagon dot under each heel, but while Indy is hanging off the tank's side mount gun the soles of Vogel's boots are different.
Continuity mistake: In the beginning where Indy is young, he fights some guys on that zoo train. When he's on the crate with the rhino in it, the background changes in each shot. Once there are trees behind and a split of a second later there are rocks, once there's a cliff and once a plain, and it all changes in splits of seconds as they change shots. (00:07:25)
Continuity mistake: When the Nazi Colonel discovers Henry Jones sen. aboard the zeppelin, Indy asks him for his flight ticket. In the first shot, Indy is standing right beside the colonel (the camera is behind the colonel and Indy comes in from the right), in the subsequent shot, Indy is standing behind the colonel.
Continuity mistake: Indy and co. leave their horses a way from the steps to the temple, but when we cut to a closer shot they're very close to the entrance.
Visible crew/equipment: When Indiana and the Protector of the Grail are fighting in the boat that is about to be chewed up by the giant screw/propeller, when the angle changes to the top point of view, you can see the rope pulling the front of the boat around to position the rear for destruction... (00:39:10)
Continuity mistake: When Indy gives his father water from the grail to drink, the grail glows from inside, and the gold inside it is bright and even. But when Indy pours water over the wound, the grail is dark inside and there are dark patches in the gold. Obviously, they had to take the light that made the grail glow out so it wouldn't be seen when Indy pours the water over his father.
Other mistake: While Indy and Henry are aboard the zeppelin Indy tosses Vogel out the lounge window, and when Vogel shouts up to the departing airship, in the next exterior shot we see the zeppelin airborne against a scenic view. Later, when Indy realises the zeppelin is turning around it cuts to another exterior shot of the airship, and the scenic view is identical to the earlier shot, only it's been flipped. The images of the zeppelin facing two different directions were composited onto the same scenic background.
Continuity mistake: In Berlin, Indy grabs his father's book from Elsa's trenchcoat. In the next shot, the Nazi pin on Elsa's tie has disappeared.
Continuity mistake: During the temple quake sequence towards the end of the film, Elsa fails to maintain her grip on the rising earth and falls onto tilted ground, sliding towards the chasm. At this point, Sallah is kneeling on the ground, but in the very next shot, he is now grabbing hold of Henry.
Visible crew/equipment: When Indy tries to reach the Grail his father convinces him to "Let it go" and Henry pulls his son to safety, then in the next wideshot of Indy, Henry, Sallah, and Marcus a crewmember wearing a short sleeved, white shirt and blue pants can be seen at the right side of the screen, just before it cuts to Henry's closeup. This crewmember can be seen again after the closeup of the knight. (01:57:45)
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: 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 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. This is due to the fact that this scene was filmed after principal photography had ended.
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.
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)
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.