The main difference that comes to mind as important to me, especially based off what I've learned from other people on this board about Story Now: In neo trad (and I'm drawing heavily on my experience of forum roleplaying in the early 2000s) you have a lot of control and curation of your character's story arc, you think about the type of person they are and where you want their story to go and then you make it happen-- in a TTRPG context that can mean a discussion where a scene is somewhat planned out between two players so they can both get what they want out of it, relatively speaking. In Story Now... well I'll let Brendan Conway explain:
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That thing they say not to do? In Neo-Trad, you probably won't just say it, you'll be more artful about it but you'd violate this principle up and down. You'd decide that you want to see two characters get into an argument over something because it would be fun. You'd look for chances to grow in the story, but probably in a specific way "oh hey, this is the perfect chance for Emrys to start having more respect for non-wizards." Rather than starting with a simple idea of who Emrys is, and letting the events and game mechanics completely mold where his character goes.
That might mean arranging for conversations within the party, yes and-ing each others direction by expressing interest and playing along (you see this A LOT on Critical Role) it might mean tying characters to backstories that govern the ways you expect them to grow as people, it can be as simple as asking the GM for certain opportunities so you can convey a certain aspect of your character around the table, or simply sticking to your concept and showing everyone else their character moments and how they change. So rather than seeing where the mechanics take you, you're flexibly charting the course ahead, playing to character arcs and events as they come up and seem cool. Its being in the driver's seat in a navigational sense.
It works well when the dynamic is that everyone is a fan of what the other players are trying to accomplish as both the audience who is totally here for the character's story arc, and the rest of the cast who have to help bring it to life. In terms of OSR, it would work well because the story is decentralized, so charting characters arcs is less about grand plots and more about the character's inner lives, occasional episodes where their backstory does show up, and relationships, which the players could chart together. With the dungeon, world lore and other emergent play elements acting as spice and letting players consider their character from new perspectives-- its the backdrop for their own artistry as roleplaying storytellers, essentially, a prompt to express themselves against.
This could be a place where story games and osr converge, in the sense that it takes the gm-as-player part seriously. I love products like the stygian library because running it, I make it very clear that I don't know what's going to happen. The only 'reveal' that's coming is dependent on dice rolls, and that's something none of us know in advance