Revealing mistake: When Donovan takes a drink from the false grail, he turns toward the camera and you can see the screw holding the base to the stem of the cup. (01:52:15)
Continuity mistake: When Sean Connery drops his lighter while trying to burn the ropes, the lid is flipped to the right. When he blows on it the lid then changes position slightly relative to the pattern on the rug, then when the rug catches fire the lid's suddenly flipped completely to the left. (00:56:40)
Continuity mistake: During the boat chase/fight in Venice, the rear of Indy's boat gets shot up and starts smoking and spouting flames and the engine starts chopping. He leaps onto the pursuing boat, fights until that boat is destroyed in the rotors, then leaps with the Grail protector back onto his original boat with Elsa. The boat is now running perfectly. (00:38:50 - 00:39:55)
Continuity mistake: In the motorcycle chase scene after the escape from Brunwald castle, Indy's hair is noticeably longer than it was in the castle scenes. It returns to the shorter length when they stop at the crossroads.
Continuity mistake: After Indiana and the Protector of the Grail escape the boat being chewed up, and the Protector reveals who he is to Indiana, watch his chest area... When Indiana asks "Who are you?" he is looking down at his chest (looking at the tattoo), and when we cut to Kazim you can tell he is holding his shirt open showing Indy the tattoo. Then when we cut to Indy and back to Kazim, his shirt is closed and he then opens it to show Indy the tattoo. (00:40:15)
Factual error: As Indiana Jones opens the door to Dr. Jones' house, note the light switches just inside the front door. They are the kind developed in the late 1970s and still in use today, not consistent with the film's setting of 1938. (00:22:30)
Factual error: German Fighters shown in the film are in fact Pilatus P-2 (Swiss-made aircraft built after World War II). The appropriate fighter of the period would be the Messerschmitt 109. (01:17:00)
Revealing mistake: When Indy is looking out at the gorge in front of him after spelling out God's name, there's a shot where he looks at the lion's head above him. You can see him from down the corridor, and you can notice as he touches the wall, it moves, showing it is a fake wall. (01:47:15)
Continuity mistake: In the beginning of the boat chase two bullets hit the windscreen of Indy's boat. In the next shot the location of the holes has changed.
Continuity mistake: When the zeppelin is shown turning around, the biplane that the Joneses are about to escape in is nowhere to be seen. (01:16:10)
Continuity mistake: When Indy takes the mail out of his pocket in the beginning of the film, he keeps the "Grail Diary" in his hand, but puts the letters down twice. Once before he opens the package and once after with the wrapper. (00:23:00)
Continuity mistake: As Indy swings through the window to rescue his father, he gets hit on the head by a vase. His hat is all crumpled and tipped quite far back, then in the very next shot his hat is neat and level.
Continuity mistake: When young Indy (River Phoenix) is advised by the fedora-topped henchman at the end of the chase, the cut he received while using the bullwhip in a previous scene changes angles. This is a flipped shot, note Indy's backwards shirt. (00:08:25 - 00:11:45)
Continuity mistake: When Indy and the protector of the Grail are in the boat about to be destroyed by the rotors, there are panels on the top of the boat that pop up and are destroyed when the boat is moving backwards into the rotors. In the next shot, they are intact, then in the next shot, they are destroyed again. (00:39:20)
Other mistake: When Marcus is knocked out at the library, look at his legs when he is dragged away - they are purposely crossed in a way which couldn't have happened naturally for a body being dragged - the actor did it to keep himself comfortable.
Continuity mistake: WHen Indy is hanging off the side of the tank, his face becomes completely caked with dirt. After he gets back up on to the tank, his face is relatively clean. (01:34:40)
Continuity mistake: In the boat chase, when the bad guys are using machine guns, in one shot we see Indy's boat in the middle of the canal. In the next shot (close-up) they are very close to a large pile of dirt (or wood chips) and in the next (wide) shot they are in the middle of the canal again.
Factual error: During the motor boat chase through the canals of Venice, Dr. Schneider pilots the vessel between two ships, one identified as the John W. Mackay of London. Although the Mackay was commissioned in 1922, it was stationed at Halifax, Nova Scotia for service in the North Atlantic. It was deployed for the first time to the Mediterranean in 1942 - four years after the purported time of the movie.
Factual error: All through the film, SS-Standartenführer Vogel (Michael Byrne) is seen wearing the Iron Cross in second place beside his SA Military Sports Badge. This is totally wrong. As the Iron Cross is a Military Award, it would be worn over or to the right of any other badge on the pocket. Also presuming this is a 1914-18 Iron Cross, he should then be wearing a ribbon bar over his left pocket displaying his WW1 Honor Cross Ribbon along with long service ribbons, etc. American Law specifically allows the accurate wear of uniforms in movies (see http://uscode.house.gov/uscode-cgi/fastweb.exe?getdoc+uscview+t09t12+443+8++%28%29%20%20AN (subsection f)) and there is no reason to believe (without evidence) that German Law forbids the accurate portrayal of uniforms. (00:53:20)
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.