A constant question during the run of all the Trek series is why Klingons look so much different, from "Star Trek: The Motion Picture" on, than they did in the original series. The real reason is the movies and later TV series had a better makeup budget. However, the "Star Trek: Enterprise" episodes "Affliction" and "Divergence" provide a canon answer. Klingons acquired genetically engineered human embryos left over from Earth's Eugenic Wars and used them to augment their soldiers. It worked but created a virus that threatened to annihilate the Klingon race. Dr. Phlox and a Klingon doctor found a cure, but it resulted in all Klingons becoming far more human in appearance. Sometime between these episodes and the first Trek movie, a cure was found, returning the Klingons to their present day "ridged-head" appearance.
The cast really are very good friends. At LeVar Burton's wedding in 1992, the best man was Brent Spiner and the ushers were Patrick Stewart, Jonathan Frakes and Michael Dorn. And when Brent Spiner recorded an album (Ol' Yellow Eyes is Back), the backing groups listed as The Sunspots are again the male members of the bridge crew.
Scotty visits the bridge from "his" Enterprise (from the original 1966 Star Trek series) on the Holodeck. As the original set had long since been destroyed, the captain's chair and center console used in the episode was donated by a fan who had recreated a life size version of the bridge. The turbolift alcove and one of the stations was built on-set (and re-used in the DS9 episode "Trials and Tribbleations"). Everything else was made up of footage from the original series that was looped.
Near the end of the episode, Picard and Crusher walk out of the cargo bay. As the doors close behind them, you can see Tasha Yar waving to the camera from inside the bay. Her character is killed in the next episode, but this one was filmed second. The actress was waving goodbye to all of her fans.
The transporter was first created by Gene Roddenberry in 1966 for the original Star Trek, as an easier (and cheaper) alternative to get members of the Enterprise crew onto a planet's surface, instead of having the ship land on the planet each time. The same holds true here. Even the original version was based on a similar effect in the movie Forbidden Planet.
The transporter system in the Star Trek universe uses a Heisenberg compensator. This is to counter Heisenberg's Uncertainty Principle, which pretty much eliminated the potential for transporters, given Heisenberg's assertion that it would be impossible to re-assemble all the components of an item transported.
In this episode, Picard is studying Fermat's Great Theorem, and says it has remained unsolved for 800 years. Five years after the episode was made the theorem was proven, by Andrew Wiles and Richard Taylor from Princeton University (their proof is not the same as Fermat's though, as they used modern methods Fermat did not know of). In the Star Trek universe, this was referred to in an episode of Deep Space Nine, and is considered as a subtle correction for Picard's statements.