Admiral Paris' rank pin shows three pips, but in all other appearances, before and after, he has four.
When the errant escape pod approaches the Ferengi ship, the Ferengi frantically call out to "raise shields." As the Ferengi were operating their ship within the corona of a red dwarf star, their shields would already need to have been raised or they would have been vaporized.
Reginald tells Seven that she is the only Borg to regain its humanity. He overlooks Captain Picard who was assimilated by the Borg to become Locutus in The Best of Both Worlds (1990) and regained his humanity in The Best of Both Worlds Part II (1990), an incident which Seven has discussed before.
However, the fact that Picard was assimilated by and returned from the Borg is quite probably a military secret that only command-level officers are aware of, and certainly not the Ferengi. In "Star Trek: Lower Decks", Lt. Steve Levy's conspiracy theory suspects that the Battle of Wolf 359 was an inside job, which it certainly was because Picard's knowledge was used to give the Borg an advantage.
However, the fact that Picard was assimilated by and returned from the Borg is quite probably a military secret that only command-level officers are aware of, and certainly not the Ferengi. In "Star Trek: Lower Decks", Lt. Steve Levy's conspiracy theory suspects that the Battle of Wolf 359 was an inside job, which it certainly was because Picard's knowledge was used to give the Borg an advantage.
Troi continues to express skepticism at Barclay's thoughts and insights regarding the Ferengi plan to steal Seven of Nine's nanoprobes even after he was vindicated in his belief that there was a plot to interfere with Pathfinder's transmissions and misappropriate his hologram. However, there's nothing wrong with her initially finding the Ferengi plot to be a little too outrageous considering the scale of the heist.
The Voyager crew is delighted by the Reg hologram's "impressions" of members of the senior staff. As a hologram, he can be programmed with the vocal pattern of any individual. Doing impressions of the staff would not be much of an accomplishment. However, it's how funny the impersonations are that the crew appreciate, not just how accurate they are. The Barclay hologram is poking fun at the senior staff which is comical.
After tasting "Mom's apple pie," Harry comments that it would have tasted better back in San Francisco. Harry's mother lives in South Carolina. However, Harry lives in San Francisco (or did prior to Voyager's mission) and is obviously speaking figuratively and meant it would have been nicer to have been nicer to enjoy the pie when he was back home.
The food item on Harry's plate is identified as "Mom's apple pie." The item contains a pie crust and a pile of green goo and looks more like a Key Lime pie filling. Whatever it is, it is certainly not apple pie. Even taking into account Neelix's questionable culinary skills, there is no mistaking the food item for apple pie. However, Neelix often takes liberties with traditional recipes (such as adding lots of spices to Vulcan Plomeek soup when it is supposed to be quite bland). This is obviously his latest variation.
When Voyager flies toward the screen at warp, it has obvious battle damage and is a reuse of footage from "Equinox Part II."
In the final scene, Barclay slightly stuttering as usual says to the computer, "C-computer, deactivate Broccoli program", using his old TNG nickname. The other characters don't notice, so it's clear it was meant by the actor to be an inside joke for the audience. The subtitles, however, spell his correct name "Barclay".
Harry says the transceiver isn't designed to handle photonic data; however, data is data no matter the contents. The Doctor has already been sent and received successfully in Life Line (2000), the previous Barclay story.
Counselor Troi chastises Barclay regarding how inappropriate it is that he followed his therapist on vacation. In her previous appearances, she emphasized that they were more than counselor/patient, but genuinely friends.
Commander Harkin tells the visiting school children that the Pathfinder office is where their "homework" gets done. Homework is, by definition, work that is taken home, not worked on in an official setting such as a school or office.
Admiral Paris mispronounces "Ferengi" and "Ferengay."
Tom Paris tells Harry Kim that they "can't go another month without mail." The ship had gone nearly 6 years without "mail," so missing a month is not that long a wait.