In The Big Goodbye (1988), Picard coughed immediately after lighting a cigarette - a normal reaction for a non-smoker. This time he puffs away without any problem. This may merely be an indication that, after playing Dixon Hill in several (not shown) adventures, Picard has either gotten used to smoking, or at least learned how to puff a cigarette without inhaling the smoke, thus avoiding the coughing. It's also possible that the holodeck has been configured to produce smoke that's only visual and not irritating to Picard.
When Dixon Hill enters his office and turns on his radio, it immediately begins playing. Radios of that time would have needed time to warm up before playing.
However, it is commonplace for TV shows and films to ignore this, for the sake of pace. The holosuite program would probably be programmed to do the same, for the convenience of participants.
If Lwaxana is a powerful telepath, then there is no need for Deanna Troi to have a conversation in the hallway when she and her mother could communicate by thought. She simply prefers this method of communication at times.
The bartender's reference to a planned German invasion of England dates the "reset" Dixon Hill scenario to the summer of 1940. However, the music playing in the background is 'Moonlight Serenade', which was not released until 1944. After this the background song is 'How High the Moon', which was not released until 1953.
The above are incorrect. 'Moonlight Serenade' (based on a 1935 tune) was first released by the Glenn Miller Orchestra in May 1939. 'How High the Moon' was first recorded on February 7, 1940, and first performed in a show ('Two for the Show') on February, 1940. The version used here is an arrangement by series composer Dennis McCarthy.
When Mr Homn drinks the flask of blue liquid, an annulus of blue fluid around the base of the flask is visible. This device allows him to apparently empty the flask in one swig, but since the flask is not really full, the amount of liquid consumed is small.
After Picard has paused the Holodeck, the "frozen" Scarface (the second villain in the scenario) can be seen to tremble.
Around the 42 minute mark, as the U.S.S Enterprise is approaching Pacifica, you can see this is a flipped shot as the registry is back to front.
In the teaser, a small vessel transports Lwaxana Troi to the Enterprise. The normal course of action would be to bring the vessel into one of the shuttle bays, specially given facts that she "hates" the sensations provoked by beaming up and that she "will never completely trust this [transporter] device." Inexplicably, she is beamed up from the small vessel instead.
Reflection at about 35:25 in side panel behind Ambassador Troi.
Towards the end, Deanna stumbles and nearly falls while she, Mr. Homn and Lwaxana are leaving the turbo-lift. She is trying to avoid stepping on the train of Lwaxana's dress.
At 25:00, Slade Bender says that "They found Alva face down in the river." As San Francisco has no local river, but has a rather large bay, he would correctly say, "Face down in the bay."