Factual error: The stamps on the mail at Sr. Jones house are Texas Statehood issued in 1945. A bit late for the scene. (00:24:15)
Continuity mistake: Onboard the ship at the end of the prologue, after Indy grabs the cross before it falls over the side, the guy in the white suit climbing up the steps. He's only reaching the top as the shot changes. In the next shot, he's already a few steps away from the top of the steps, telling his men to stop Indy.
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.
Continuity mistake: When the Hatay soldier is decapitated in an attempt to navigate the temple's defences, his head rolls along the ground and Elsa turns her head around and looks over her shoulder. In the next shot, she has now turned her full body around, and in the shot after that, she is looking over her shoulder again.
Continuity mistake: When Elsa's hand slips from its glove whilst Indy is holding on to her, Elsa falls and she is holding the glove in her right hand, but when she falls into the chasm, the glove has gone.
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)
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)
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: 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)
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: During the boat chase, after the one boat blows up, another comes up behind Jones and Elsa. Jones starts turning the boat. In the next shot, the man in the boat behind them is shooting at them. They're not turning, they're going straight.
Continuity mistake: Before they make the second fighter pilot crash, Henry hands his bag to Indy. In that shot, the handle of Henry's umbrella is closer to Indy's left shoulder. But, in the next shot from in front of Indy, the handle is closer to his right shoulder. Then, when Henry pulls the umbrella out, the handle is back near Indy's left shoulder.
Continuity mistake: In the cave, young Indy is giving instructions, his hair is down both sides of his face. When he removes the snake saying "It's only a snake", his hair style is changed.
Continuity mistake: The gap between Indy and the chasing motorbikes changes considerably between shots, close to him than far apart the close again and at the checkpoint out of sight.
Continuity mistake: The dirt road the solo motorbike rider takes heading towards Indy is completely different to the road Indy uses just after he knocks the solo rider off.
Continuity mistake: When Indy is on the boat and says "Too small for two of us!", it pans down and you can see the cross halfway out of Indy's pouch. Both of the arms and head are on the outside. Only the bottom half is in his pouch. The bad guy then proceeds to open the pouch, revealing only the top before taking it out.
Continuity mistake: When they first meet, Walter Donovan hands Indiana the glass of champagne almost filled to the brim. The next shot shows Indy's glass with much less champagne in it.
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.