Bowdlerize: While both versions are up, the version of the "I NEED U" music video that was first released (the shorter one, not the Original version) cuts many key scenes both for length and for violent content (which includes depictions of murder, suicide and overdosing)... so you just get scenes of all the characters being sad, pretty much. This means that you have to watch the Original version in order to understand the story.
The Cast Showoff: Both Suga's piano skills and J-Hope and Jimin's dancing (aside from the choreography seen in music videos) are incorporated into the story. Suga appears playing the piano in several scenes, including "First Love", "Blood, Sweat and Tears" (though in that one it was a pipe organ) and the Highlight Reels, with the piano being a key plot element. Hoseok and Jimin dance in a studio in-universe, which is shown in the Highlight Reels.
The Danza: The seven main characters share the members' real names.
The Prologue video was deleted and re-uploaded by BigHit, this time with changes to the credits music and with the absence of an additional post-credits scene. This Stinger showed Jin in the truck on the beach, looking at the polaroid picture he took with Yoongi at the gas station, and noticing that Yoongi has disappeared from the photo.
It also shows that the scene where Jungkook is standing on top of a building had extra few seconds at the end, where he jumps.
Fan Nickname: Since the series wasn't explicitly given a name until 2018 (which is the very vague "BU"), the story has been referred to by fans as the HYYH story/storyline or the HYYH series, as the story officially started during the "The Most Beautiful Moment in Life" album series; there's also the fact that the Notes are named the HYYH Notes (after the romanization of the original Korean title of The Most Beautiful Moment in Life, "Hwa Yang Yeon Hwa"). After the name "BU" was established, fans have called it the BTS Universe, the Bangtan Universe or simply the BUniverse, but "the HYYH series" still sticks with some fans.
Meaningful Release Date: The release of the last chapter of SAVE ME was on April 11th of 2019 (Korean time); not only is April 11th both the day Jin wakes up whenever he goes back in time, but it was one day before the release of BTS' album MAP OF THE SOUL: PERSONA. It also happens to be the date of release of Singin' in the Rain, a film heavily referenced in the "Boy With Luv" music video, also from PERSONA.
Meta Casting: The characters are inspired and named after BTS' members, with the members themselves having input in the story, so (to a degree) they share some key elements and personality traits with their real-life counterparts, such as the The Cast Show Off examples above. RM has said that the group sits down to discuss the story of each video with their label, trying not to stray too far from their real-life personalities.
Missing Trailer Scene: Some of the teasers for BTS' music videos contain original scenes that are not used in the music video itself, but are not integral to the storyline, like in the "RUN" teaser (featuring the boys staring at the camera in front of a black background). In the case of the first "FAKE LOVE" teaser, however, it centres almost entirely on a scene that doesn't appear in the music video: the boys entering the "Magic Shop" (preceded by a Dictionary Opening for the "Magic Shop" itself), and exchanging key objects seen in the WINGS teasers (a Snickers bar, a mirror piece, etc.) with objects seen in the Highlight Reels (a cake, a hair tie), with Jungkook then getting pushed into the room, leading to the final scene of the music video.
No Budget: The "I NEED U" music video, filmed while BTS and their company were going through one of their most financially difficult periods. It features people from staff as actors (the two guys Jungkook bumps into on the street are actually managers), and the car Namjoon puts gas into is the music video director's.
Stunt Double: The scene where Taehyung jumps off the tower in "Prologue" was filmed with a stunt double.
Teasing Creator: Try asking the members about the story. You'll never get anything you don't already know. They happen to enjoy seeing the fan theories (and yes, at least one of them has tweeted a post laughing about it).
In the (previously mentioned) interview for Rolling Stone India, RM said he once saw a theory video that got very close to the real deal. He didn't say which.
It's also unclear how much they themselves know about the story, or at least about the symbolism (see Meta Casting above). Their reaction video to "RUN" has them say stuff like "we don't really know what this means...", though it's debatable how true that is.