When Barclay goes to the holodeck to track down the problems, the door opens automatically, as do doors to crew quarters, turbolifts, etc. The holodeck is accessed by communicating through the computer panel next to the doors.
When Picard warns Moriarty about criminal activity, Moriarty blames his creator by saying his crimes were "the scribblings of an Englishman dead now for four centuries." Arthur Conan Doyle was Scottish, not English.
As the merger of the two planets draws closer, Geordi claims that "the gravity well is beginning to intensify" and Riker voices concerns about the Enterprise being pulled in since she might be too close. Since the gravity exerted by the any object is directly proportional to its mass, the combined gravity of the two planets would remain the same. If anything, it would be diminishing after the merger since that will ignite fusion, converting mass into energy which is radiated away.
The transporter console reads "Transport log 721." Instead of reading it, Data asks the computer what is being displayed here. Computer reports that it's transport log 759. Data says, "That is the correct log."
In the first scene, when La Forge tells the computer to "freeze program," the fire in the hearth keeps burning and flickering.
As Barclay enters the holodeck, just after Data and Geordi exit it, visible creases reveal the floor is a black carpet or cloth with yellow tape on it.
Moriarty's shirt clearly buttons all the way down the front. Those styles of shirts didn't become popular until the 1920s well after the setting of the various Sherlock Holmes novels.
Both Moriarty and the Countess have capped teeth. Veneers for teeth were not available until the 1950s, which does not suit the 1890s setting of the holographic novel, nor the show's main setting in the 24th century when this would be obsolete.
Riker and La Forge speak to Moriarty on the view screen. At the end of the conversation Riker asks La Forge to start working on the problem, La Forge replies "Aye Sir" but his mouth movements don't match the sound.