Continuity mistake: When Data and Troi are scanning the list of elements, searching for something to cause an explosion, the list is scrolling "up" the screen. When she tells Data to stop and go back, he does, but the list continues to scroll "up" instead of back "down".
Continuity mistake: In the scene where Data is brushing Timothy's hair into a copy of his own hairstyle, he stops, kneels down and asks Timothy "Timothy, are you having bad dreams?" In this shot there is a big tuft of hair on the right side of Timothy's head that sticks straight out. When their conversation is over and Data stands up in the wider shot, all of Timothy's hair is in place.
Visible crew/equipment: In the opening scene when Data is portraying a character from Shakespeare's play "The Tempest" a small card can be seen attached to the back of the book he is carrying (likely containing his lines), but after increasing the amount of torch light the card is no longer there.
Plot hole: When Picard, Geordi, and Troi encounter one of the first time disturbances it results in one of the runabout's nacelles using up all of its fuel. Data says this is due to it having been in operation for 47 days (according to the plasma conversion sensor). After this the Captain reaches for the bowl of rotting fruit which causes him to scream in pain. The other crew members rush in and Troi scans his hand. She tells Picard that his hand is metabolizing at approximately 50 times the normal rate. Data and Geordi then discover that the temporal disturbance which covers the fruit also covers the nacelle that has lost all of its fuel. Data also notes the disturbance extends outward from the hull, about 17 meters from the ship and is spherical in shape. The problem here is that according to what Data said earlier the engine was active for 47 days even though it had only very briefly come into contact with the time distortion - maybe 1 second at most, but likely far less time than this since the ship was at warp when the fuel was consumed. So if we assume the engine was in contact with the fragment for 1 second then time is actually moving at 4,060,800 times the normal rate - not 47 or 50 times normal.
Continuity mistake: Near the end of the episode, right after Lore fires the phaser at Dr. Crusher, Data tackles Lore and knocks the phaser out of Lore's hand. The phaser flies towards the cargo transporter and when it impacts, it breaks in two. You can see a piece of it go flying off to the left. Then shortly after when Data throws Lore onto the transporter pad and Lore picks up the phaser, it's in one piece again.
Continuity mistake: In the beginning of the episode, Riker, Geordi, Worf, Data, and Pulaski are playing poker. Pulaski puts two neat stacks of chips in the center beside some chips in a pile. It then shows Worf putting similar stacks in the center but Pulaski's stacks are not visible and they should be. Then when Worf wins the hand, the camera is showing a wide shot of him pulling the chips towards him and you can see his and Pulaski's neat stacks of chips.
Other mistake: When Scotty goes to the holodeck to visit the original Enterprise, the computer tells him there are 5 Federation vessels with that name. (NX-01 served before the founding of the Federation so this is technically correct.) Scotty specifies NCC-1701, "no A, B, C or D." The computer then tells him the program is ready. The computer should have asked him if he wanted pre or post refit NCC-1701. The ship we see in the Original Series was rebuilt/refit into the ship we see in The Motion Picture.