Plot hole: About two thirds of the way through this episode, Data deduces how Moriarty was able to 'leave' the holodeck. The big reveal is that he never did leave the holodeck, he merely reprogrammed it to simulate the rest of the ship without Picard, Barclay or Data's knowledge. While it is believable that this would fool humans like Picard and Barclay, it is ludicrous to suggest that Data would be taken in by it, even for a second. Data is an android whose perceptions of sight, sound and the world around him are far more sophisticated than humans. To list what we know of Data's perceptive abilities from previous episodes would take up the entire page, but suffice it to say he should have immediately recognised the 'Enterprise' as force fields and holograms rather than the genuine article. Note: this goes beyond a 'character mistake' or anything like that. Data's enthusiasm for Sherlock Holmes style deduction should have led him to eliminate the impossible before considering what was probable. Data accepts the impossibility of a holodeck character existing is the real world before discounting the possibility that it was still a holodeck simulation. This contradicts an awful lot of what is known about Data's abilities and powers of deduction. Although it serves for a good mystery the fact is it takes Data far longer than it should to deduce, or even guess at, the truth of their situation.
Continuity mistake: In one scene on the planet, Riker's clone is lying under a table fixing something and Riker is sitting in a chair talking to him. There are two camera angles used. In one angle when the clone is talking and a body double is sitting in the chair he is facing directly towards the table but in the other camera angle where you can only see Riker, the chair is at an angle facing towards where his clone is lying on the floor.
Continuity mistake: William Riker is playing cards with La Forge and Worf. The camera is focusing on Riker and there is no 4th chair. When Tom Riker enters, William says "Why don't you join us" and now removes his trombone from the 4th chair which has suddenly appeared. (00:31:25)
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.