Plot hole: The entire premise of the episode is contradictory; the franchise makes abundantly clear through the so called Prime Directive that the Federation abides to a code (often creating moral dilemmas that may require to stretch the rules) that says that their staff is not supposed to interfere in world that haven't reached warp capabilities, nor involve themselves in their internal matters. In the first half of the episode, Riker and the others just go 'mingle' with the frisky natives without a care in the world, and yet in the second half the Prime Directive itself is referenced explicitly and it is part of the plot.
Suggested correction: They are just enjoying the hospitality of the planet's inhabitants. They are not interfering in their internal affairs or the development of their species. It's only when Wesley gets into trouble that the Prime Directive comes into play. The entire mission of The Enterprise is to make contact with other planets.
Even though they are supposed to make contact with other planets, it's pretty clear in the rest of the series after the first few episodes of Season 1 that they do not (intentionally) contact people that have no warp capability.
Plot hole: The binars needed two people to upload the data back to their world but they only intended on keeping Riker in the holodeck. Riker is just one person. Picard came into the holodeck purely by accident - even Minuet said this was true. If Picard hadn't been there, Riker could not have uploaded the data on his own and the binars' plan would have failed.
Suggested correction: The Binars only trapped Riker because they thought they might need someone to restore the computer for them, which wouldn't have taken 2 Enterprise crew members. It was the fact that the Binars got to their planet too late that required Picard and Riker to work together. However, the Binars already knew Picard was on the Holodeck before they initiated their plan because when Picard exits his ready room he says he's going to join Riker. Had Picard not done this, certainly the Binars could have kept a 2nd crew member on-board somehow, if needed.
Conspiracy - S1-E25
Revealing mistake: After the fight with Riker, Quinn throws Worf through the room, hitting a wall. The replicator part of the wall moves.
Revealing mistake: In an overhead shot of the Brattain, it's clear that the ship's (misspelled) name is just a poorly applied decal.
Visible crew/equipment: While Picard talks to Rasmussen in his ready room, when Rasmussen says, "I can't get quite as worked up as you," it cuts to a wide shot, and there's a white tape mark (T-mark) taped to the floor under the desk. (00:32:45)
Visible crew/equipment: On Romulus, when Picard and Data are told, "Eat your soup, courtesy of a loyal establishment," the camera turns to face the duo dressed as Romulans, and an actor's T-mark is visible on the ground behind Data. (00:42:40)
Visible crew/equipment: Boom action is visible in reflections on computer consoles in the back, while Picard and LaForge talk to each other on the bridge. (00:40:15 - 00:40:46)
Continuity mistake: When Data goes into his laughing fit, Troi is at her regular seat, but when he stops and remarks it was a wonderful feeling, the wide shot of the crew shows her seat empty. But then, when Picard is hailed, when the moon returns to its orbit, Troi is back at her seat.
Continuity mistake: When Lore backhands Worf in the turbolift, his eyes close but open back up as he is sinking down. Then, in the close-up, his eyes are closed again.
Visible crew/equipment: After the intro, when Garvin tells Gia to go home, Data is standing on the stairs and the actor's green mark is somewhat visible at Data's feet. This green T-mark is clearer when Data hands Garvin the radioactive box he's carrying. Then, when Garvin walks over to the table, there's an orange T-mark on the ground. (00:04:55 - 00:07:40)
Visible crew/equipment: After Gia tells Data to remove his hood, he asks how long it will be until Skoran and the others return. Then, when Data tells her that he must hurry he takes a step forward, and a green tape mark is visible on the floor. (00:36:35)
Visible crew/equipment: While Troi tells Riker about the Captain's grueling negotiations between the Gemarians and Dachlyds, when Picard joins them on the bridge, there are three visible T-marks taped to the floor. (00:01:50)
Other mistake: During his final discussion with Tomalok, Picard tells him that the Enterprise will escort the Romulan ship back to The Neutral Zone. However, as the episode ends, the two ships are heading off in opposite directions. Both the Enterprise and Romulan warbirds have demonstrated in the past that they can spin on their axis from a standstill. There is no reason for either ship to head away from the other to turn around.
Conspiracy - S1-E25
Continuity mistake: Lt. Commander Remmick has three solid pips on his uniform when Admiral Quinn is beamed to the Enterprise, implying his rank to be Commander rather than Lt. Commander. In all other shots in the episode, his rank is displayed as Lt. Commander. (00:25:11)
Audio problem: After the offer of refreshments has been declined, Harmony says "I have been expecting you", but her mouth doesn't move.
Continuity mistake: Near the end, after being subjected to the snowy scene, only Mr. Leech has snowflakes on him. They're on him as he steps into the corridor, then they're suddenly gone.
Deliberate mistake: To fix the error of the Enterprise firing phasers from the torpedo bay, the Blu-ray version uses footage from "The Best of Both Worlds" to show phasers firing from the dorsal emitter. In doing so, they created a new goof; the planet El Adrel is not visible in the background during these shots.
Suggested correction: I rewatched the original version on the old DVDs. El Adrel wasn't visible in the original either, so this is not a new error.
Audio problem: Near the end, when Leech points his gun at Crusher, there is the sound of him cocking the weapon. However, when we see his gun immediately thereafter, the hammer is in the uncocked position.
Visible crew/equipment: In Wesley's quarters, when the camera looks at Geordi, there appears to either be a person reflected on Wesley's device or a boom mike shadow plays across it.
