The story, unsurprisingly, is absolutely ridiculous. The characters are summoned to Club Velvet, a dream-like world conjured by a mysterious woman called Elizabeth. (In the original game, she helps the main protagonist fuse powerful beings, called Persona, in the Velvet Room.) She explains that they're asleep and, through the power of their heart, memories and imagination, can move like professional dancers. The cast are confused but ultimately agree to let off some steam before returning to the real world and continuing their fight against the vicious Shadows.
I didn't mind this setup. It's a dancing game, after all.
Dancing in Moonlight takes place during a vaguely-defined night towards the end of the original game. Characters will occasionally mention the wild year they've had together, but otherwise it's impossible to know when the game is actually set. That's by design, of course -- Dancing in Moonlight is a side-story that doesn't, and without serious retconning can't, have any impact on Persona 3. It's a shame, but the plot has to be fairly trivial.
The game has no way of knowing if you played Persona 3 or which relationships you focused on.
The narrative no-man's land has a huge impact on the characters, however. The game has no way of knowing if you played Persona 3 or which relationships you focused on. (The core series has a unique calendar system that forces you to pick who, and how you spend your free time.) That means everyone has to be written in the broadest way possible, focusing on their core personality traits and none of the development that was delivered through their 10-part backstory, or 'Social Links', in the original game.
Dancing in Moonlight does offer 'Social Events' that unlock as you complete tracks and meet character-specific conditions. The vast majority of these scenes are forgettable, though, and reveal little that wasn't already covered in Persona 3. The characters are told early on, too, that their memories of Club Velvet will disappear once they awake. Most of the heroes then question the point of doing anything at all, beyond simply dancing and enjoying themselves, if their hard work will be undone afterwards.
Watching these fun but shallow scenes left me feeling hollow. I had experienced so much with these characters and forged deep, believable relationships with them in the original game. As I listened to them chat, I couldn't help but think: Don't we know each other better than this? Wasn't our friendship, and in some cases romance, more than small talk?