LeVar Burton was allowed to regrow his beard because he needed it for his wedding. But unlike The Outcast (1992), it didn't just appear and disappear, it was written into the episode. It's unknown whether or not he shaved the beard off in future episodes because he lost his bet with Doctor Crusher.
When Data explains to Captain Picard his decision on defending the life of the exocomps, he mentions that when his own life was on trial, Picard defended him. This is a reference to The Measure of a Man (1989) in the second season.
The Exocomps were anthropomorphized as the story passed through several scriptwriters. In Naren Shankar's script, they were changed to look like mobile shoebox units. But they just wound up looking like footballs wearing trainers (sneakers), according to the design department. When Jonathan Frakes elected to direct, they gained little dips and hesitations in movement, rather than just R2D2-like beeps and wheezes to denote emotion and thought.
The original title was "The Underground Circuit" by L.J. Scott. The final teleplay was credited to Naren Shankar as his first script for the show.
Some Scifi purists thought the story would have been better if the Exocomps had stayed as boxes with no discernible affectations as originally planned. But what made the episode so popular was the way they emerged in the final result. It allowed an audience to identify with them more fully. L.J. Scott's initial script also drew parallels to similar situations in Earth's history (the episode does remind somewhat of Short Circuit (1986), the story of a robot that develops life of its own). The original ending would have been to come up with a method to know if the Enterprise computer could become self-aware.