While Firefly, it still left some unanswered questions with regard to what happened between the two. The more condensed nature of film storytelling, however, necessitated that certain mysteries would be ed over, or only addressed obliquely, and among those is the question of what the crew (and their nemesis, the Alliance) were up to in the intervening months between Firefly's last episode and Serenity's opening minutes?
The most significant discrepancy between the film and the show is the absence of the regular characters Inara Serra and Shepherd Book. Inara tells Mal point-blank in the episode "Heart of Gold" that she’s leaving Serenity, and at the beginning of the film, she’s no longer on the ship. Book is a more complicated affair. They visit him on the planet Haven and Book and Mal even have an extended heart to hear without ever mentioning why he was there and no longer on Serenity. Unlike Inara, there was no indication Book was leaving in the show, and the film's focus on its own story leaves that question unanswered.
Since the film's release, however, a veritable cottage industry of comic books and novels sprang up resolving these unanswered questions, from the mystery of Book's incongruous backstory to Mal's dark secret from the Battle of Serenity. The midquel comic, Serenity: Those Left Behind, in particular wraps up plot threads from the show not addressed in the film, as well as providing set up for the major events of that film. For instance, Lawrence Dobson, the villain from the pilot episode, is revealed to be alive and seeks his revenge against Mal and the crew, the Hands of Blue, make one last effort to apprehend the Tams before they're incinerated by Serenity's engine fire, which then sets up the Operative as the one tasked with tracking down Simon and River.
Further, the comic finally explains why Inara and Book are no longer on the ship. With Inara, fans had an idea that she really left due to the unresolved romantic tension between herself and Mal, both of which are confirmed by the comic. With Book, fans finally got an actual answer. After Book aids the crew of the Serenity in a heist, Book and Mal get into an argument over Mal going against his word to take Inara back to fulfill her “duties”, ending with Book punching Mal in the face when Mal implies Book’s word means nothing as well. In the end, after finally dropping off Inara, Book informs Mal of his intention to leave Serenity, fearing his interactions with Mal are corrupting him.
Whedon actually intended to address Inara's absence more explicitly, with those scenes ending up on the cutting room floor, but he never seemed to have an explanation in mind for Book. Granted, Whedon could hardly answer every question within a two-hour film, but it was enough of a gap to necessitate a comic explaining it, so his lack of any explanation regarding Book's absence is indeed curious. Even the briefest of mentions of a reason for Book's absence would have helped increase the sense of continuity between Firefly and Serenity. Considering some of the noted discrepancies between the two (like the discrepancies between River's escape in the show and the film), more explicitly addressing the more glaring disruption of continuity, in Book's explained absence, might have helped preserve a greater sense of continuity despite such differences.
So while Serenity does answer some questions, it leaves more on the back burner, allowing fans to fill in some of the holes for themselves, or letting Firefly's companion comic books take on the load the film could not. Doing so allowed the film to stand on its own, as its own independent story, and thus bring the stories of the Serenity and its crew to a larger audience.