When Captain Picard is talking with Groundskeeper Boothby, Picard's breathing is clearly visible, and fog can also be seen in the background, but Boothby's breath cannot be seen. Most likely the wide shot of the scene, and Patrick Stewart's close-ups, were shot early in the morning. Then Ray Walston's close-up was shot later in the day, when it was warmer, so his breath was no longer visible.
When Wesley and Nicholas are seated at his desk in his quarters, the position of Wesley's left hand changes between shots.
In the explanation of the maneuvers, the term "low apogee" is used to describe a trajectory around Titan, a moon of Saturn. In the strict sense, an apogee is the highest point in the orbit of a satellite around earth. Even when accepting that the term may also be used for other planets, the fact remains that the term is used here to refer to the orbit of a ship around a moon, not a planet, hence it is not correct. A correct term would have been "low apoapsis".
While being questioned at the hearing Wesley refers to the female Admiral as 'Sir' when the correct military reference would be 'Ma'am'.
A banned maneuver is a banned maneuver, meaning Locarno's academy career would have ended in a reprimand instead of a blaze of glory.
In the final moments, as Wesley is sitting on a bench on the grounds of Starfleet. Captain Picard walks up to him. Wesley continues to sit and speak casually with him. A cadet at any service academy would immediately spring to attention if any officer approached him, and the first words out of his mouth would be "Sir."