It seems at a high level.
Old School - Game driven, what is External to the PCs is more important. "The main character of the World of Warcraft, is Azeroth."
New School - Story driven, what is Internal to the PCs is more important. "The story is about me, and how the adventure relates to my character."
Everything else seems to fall into either camp to me.
My main problem with the latter analysis is that an emphasis on teamwork and mutual understanding is pretty much essential in every "new school" group I've ever participated in. That's why we have things like Session Zero, the X and O cards, lines and veils, Dungeon World's bonds, etc.
You need only look at the big-name (and, heck, even
small-name) 5e D&D game podcasts like Critical Role and The Adventure Zone to see this stuff. It's absolutely NOT about five (or whatever) individual, deeply selfish people who exclusively care about their own story and NOTHING else. It's about five
teammates who are working together on both their communal goals, and on their personal stories, both in how those stories work individually and how they braid together, how the characters change one another.
Part of why, when I was asked earlier (I presume incredulously) whether it was like being a method actor, I answered simply, "Yes." Because that
is what it's like. The character
is a role I'm stepping into, immersing myself in, wearing almost like a second skin. I need to have the what-it's-like to be that person. Jumping to an entirely different character, I lose that, and have to rebuild it--which takes a
long time. Weeks, perhaps months.
So I guess I'd phrase it as, "The story is about
us, and how the adventure relates to
our characters." Old school in-character play is like a genealogical or anthology story, where we follow the torch as it passes from bearer to bearer; the individual bearers are far less relevant than the torch itself. New school in-character play is like a TV show with a core cast, where we get deeply invested in
these characters and what they're doing.
It's the difference between
Foundation (or at least the first book, with its many shorter stories) and
Babylon 5.