Here’s an attempt at a slightly less tendentious version of the contrast, though I also prefer the first in most instances:
Version 1:
1) GM gives canned description of room, including telegraphing a trap.
2a) Players stupidly ignore the telegraphed clue, triggering it when they interact with it, or....
2b) Players asks questions about what they can see of the trap, GM answers, in loop.
3) Players describe an attempt to solve trap (or leave it alone); GM adjudicates success. (There may be specific rules for this or she may rely on "obviously succeeds, obviously fails, or flip a coin," but a plan that succeeds by common sense should succeed without requiring a roll.)
Version 2:
1) GM gives canned description of room, with or without telegraphed description of trap.
2) GM asks for perception checks, or players declare that they are searching for traps. Either triggers a roll based on character skill, resulting in...
3a) A party that doesn't notice the trap, likely triggering it, or
3b) A party that notices the trap, and GM description of the trap.
4) Players attempt to solve trap (or leave it alone); if they attempt, a relevant skill roll will be required, and this will determine whether the trap is disabled or goes off.
Version one (OSR) grounds players more directly in the fiction, forcing them to engage with it; version two ("modern") allows for more mechanical differentiation and expression of characters (which many people in this thread have cited for why they don't like OSR play!)
N.B. older versions of D&D do indeed feature rolling for traps and secret doors; games designed with contemporary OSR ideals (which, despite the acronym, really is a newer phenomenon) tend to not have this. FKR takes this in a more radical direction still, basically removing all rules abstractions. OSR is a playstyle not a rulesset, albeit a playstyle that's easier to implement with some rules than others!
N.B. there's also a whole spectrum in what gets abstracted to what degree. A very large number of tables resolve social encounters via 1 and combat encounters via 2, though of course there's no reason you couldn't flip that.