For me, it comes down to a few specific areas: ease, emphasis, and personal nostalgia.
BECMI is easy to run. The game's built-in tools make world-building (both dungeon and wilderness design) super easy. Once you have a handle on the core mechanics (searches, turns, wandering monster checks, reaction rolls, traps, combat, wilderness travel, etc.), the game runs smooth as butter. It's structured enough that I don't have to grope to make rulings all the time, but never so overcomplicated that I have to spend time looking up fiddly little rules. I can prep a world in a week, but then run it for a year.
As a "forever Referee," I like the old-school style's emphasis on challenge, exploration, world-simulation (i.e. playing to find out what happens rather than playing through a storyline that I, as DM, already know the ending to), and the concept of the "Grand Campaign" (the idea that instead of a single ensemble "cast of characters" all moving through a plotline together, the campaign instead consists of a large number of atomized, semi-independent adventurers who all have their own individual goals and aspirations, and the game is about finding out which of these characters will "rise to the top" and become movers and shakers, builders of nations, maybe even Immortals — as well as a grand struggle fought along factional lines between Law and Chaos). I don't mean to claim badwrongfun or yuck anyone's yum when I say that trad campaigns (the kind with a fixed cast and a storyline) are simply too restrictive and predictable for me — been there, done that, it's a solved game that doesn't interest me anymore.
Finally, I can't deny that nostalgia is a part of it. I'm a Millennial (to the extent that generational cohorts mean anything). I first played D&D on the 1070 black box, but I really got into the game as a high school freshman, first using the 1106 Classic set and then quickly moving over to AD&D. (While we didn't mix Classic and Advanced, heeding the strictures in the text that they were two separate games, we enthusiastically ran 2e games that included whatever 1e materials we could get our hands on. Even though we were definitely playing 2e, the 1e PHB and OA were always essential components.) We played AD&D in a very trad style, because that reflected our sources of inspiration (Tolkien, video games) and because it was in the zeitgeist at the time: the 90s were the days of "Vampire is for the real roleplayers!" vs. "Nuh-uh, AD&D can be for real roleplayers too! It's only a few bad apples (munchkins and grognards) doing all the hack-n-slash, rollplaying, and metagaming!" So to have the OSR come along later and induce me to finally go back and give those old rules that I started on a fresh look, and to discover that they were meant for something wholly different — an old-school Grand Campaign is so different in purpose and character from a trad campaign that I now consider them to be part of functionally separate hobbies — it was more than just enlightening, it was mind-blowing.
Now as to the OSR specifically, while I don't always go along with the ways that it revises history or defines itself against the modern game by emphasizing things like lethality or interrogating the fiction in lieu of using dice mechanics, I do think that there's one aspect of the OSR which has been undeniably valuable: the OSR has divested the game-engine underlying OD&D (in the broad sense — LBB, BX, BECMI, etc.) from the fantasy milieu itself and repurposed it for every genre, from modern to sci-fi to supers. There's a Swords & Wizardry derived, OD&D compatible OSR game out there for every genre, which makes OD&D now a universal system as adaptable as GURPS or Savage Worlds. That's phenomenal. It means that as long as I'm happy with the mode of gameplay that OD&D does best (challenge-based exploration that emphasizes the world over the characters exploring it), I never have to leave the OD&D ecosystem just because I want to play a game that isn't set in pseudo-medieval fantasy-land. If I want space opera, there's White Star. If I want post-apocalyptic, there's Mutant Future. If I want four-color superheroes, there's Guardians. And so on. I can't say enough good about that fact.
