One thing I've enjoyed in the OSR is how it's allowed fresh looks at the games many people first picked up in the 1970s and 1980s, taking those games in directions that were, at best, rarely explored in their original era.
Probably the innovation I've enjoyed the most is the emergence of products to support sandbox play. It's one thing to advise a GM to let the players wander the "map" freely. However, that isn't always all that easy in practice. Enter a number of products designed to support free ranging players. Companies like Sine Nomine Publishing, Lamentations of the Flame Princess, Autarch, and many others have all released products designed to give a GM tools for such adventures. Now that I think of it, this has even entered more "modern" style games. For example, The One Ring and Adventures in Middle Earth both have tools for wilderness adventure that have that old school hex crawl feel to them.
Old school D&D style games have also been taken to genres they had rarely, if ever, been taken to. Some flavors of the OSR have embraced more of a weird fantasy feel. Others dive into swords and sorcery. Games like Dungeon Crawl Classics and Astonishing Swordsmen & Sorcerers of Hyperborea have felt comfortable at adding laser rifles right next to +1 swords. TSR made some efforts in the 1st and 2nd edition eras to fit D&D to other settings - Lankhmar is one such example as are the green historical references of AD&D 2nd edition. However in both cases it was a bit of a rough fit, whereas OGL-based games allow for a nice mixing and matching of the proper tools for a given genre.
The genres the OSR has supported are well beyond what we thought D&D could support back in the day. Stars Without Number and White Star have given a nice take on science fiction gaming. There are a number of games designed to emulate superheroes within the OSR. I'd've been shocked in 1983 at the idea of such a game.
This isn't to say the OSR is the one true way. My favorite game of all time is that living fossil, Call of Cthulhu. I've quite a bit of fun running a Star Wars game with Fate Accelerated. I'm going to try for a Golden Age Champions game at some point. And Cthulhu Dark is screaming to be tried at some point. An OSR horror game would feel very different from Call of Cthulhu which is different from Trail of Cthulhu and Cthulhu Dark. And at any given time I could be up for any or all of them.