One of the huge differences between (say) Dungeon World/Blades “Sandbox Story Now” play and Trad Sandbox play is the systematized avenues for player aggression to advocate for their PCs (and through that advocation, wrest control of play trajectory).
A related thought:
Does the GM, in narrating consequences,
extrapolate from what s/he has already prepped or imagined about the gameworld? Or does the GM, in narrating consequences,
have regard to the players' evinced desires about the trajectory of their PCs?
(Of course, this also relates back to the discussion upthread of protagonism.)
The first approach is
neutral GMing. I think this is pretty typical of classic/trad sandboxing.
The second approach is one I associate first and foremost with Burning Wheel, but I've used it in other RPGs too: 4e D&D, Prince Valiant, Cthulhu Dark, and to some extent Classic Traveller.
Upthread I gave a brief account of the episode of play, in our Prince Valiant game, in which the squire PC was knighted by an NPC as the result of an attempt to just ride past him after he refused to joust with a mere squire. In a "neutral" approach the GM would consider the personality of the NPC, the customs of knighthood, etc and extrapolate a "realistic" likelihood of the NPC knighting the PC. But in our game I simply called for a Presence vs Presence check. This keeps the focus on
what is at stake for the character in the scene rather than
how often do squires get knighted by proud knights so as to create fitting opposition for those proud knights. I think only the second would count as
exploring the GM's world.
In both cases, the world of course is living and breathing. That description doesn't at all capture the difference between the two approaches to adjudication.