Who likes lots of caveats? Because if you do, I have a great post for you.
Caveat the first - I'm slightly misquoting John Green here, but much of what we assume to be binary is in fact spectral. There is no easy switch between New School and Old School, because both styles are played in numerous ways. Rather than a line, I think that D&D in particular is a tree, with a complex net of roots underneath and many branches swaying overhead, but every point on the tree is in fact the tree, and lop off the branches or tear out the roots and the tree will die. It needs both to survive.
And as a sidebar, I've been playing D&D in various forms since about 1982, and a lot of what is presented as Old School in this thread is nothing like my play experience. Which is fine, it just means I've been on a different point on the tree than other people.
Caveat the second - New School isn't particularly new at this point; we're talking about trends that have been present in the game for decades.
That said, two broad tendencies I've noticed.
New School tends to be fiction first, while Old School leans toward "play to find out." OS games don't tend to have clearly defined story arcs, they have environments and situations that the PCs react to. This is not absolute; there were some epic (and slightly silly) storylines in my games back in the Eighties, and my current 5E game is a procedurally generated sandbox in the Land of Oz.
Second, New School characters are more defined mechanically than Old School characters, while taking longer to create. OS classes are just three or four things you can do, and the main mechanical difference between two characters of the same class will be the selection of equipment and/or spells they've picked up while adventuring. This one is very D&D specific; I played some Champions in the eighties as well, and that is not a simple character creation system.
And that's really it. No point in worrying about which is better - find a branch or a root that you like, settle in, and don't throw acorns at the other inhabitants of the tree.