A little late to the party, but in my experience, this is the most salient difference:
My original group started with B/X in the 80s. After 5e came out, we came back together to play again. To this day, the players do not ask me if they can do something they want to try. They tell me what they are doing, and then it's on me to adjudicate it.
When I've played with players who started since 2000, they will generally a) ask me if they can do something, or b) decide on some skill and roll unilaterally.
In other words, the "old school" players are comfortable with what I call "DM-based resolution." The "new school" players are more comfortable with "rulebook-based resolution." They prefer hard, mechanical points-of-contact with the game. If the situation does not lend itself to a specific rule or mechanic, they want to confirm how the DM will adjudicate it before they commit to the action.
While no doubt both styles have existed throughout the game's history, I believe the rules (specifically 3.x and 4e) were explicitly designed with rulebook-based resolution in mind, while the pre-2000 rules were explicitly designed with DM-based resolution in mind, and so this is a valid distinction.
FWIW, I do not feel that either paradigm is better than the other, though I have a preference (both as DM and player) for DM-based resolution. 5e seems to me to be an effective hybrid (New Old School?) in that although the essential flow of play as described in the PHB follows DM-based resolution, between the skill system and class abilities there are enough hard points-of-contact that both "schools" are served.