Provided the Game Master in question implements and weaves plot here and there, not making the players the center of a story and letting them decide their own fate by their deeds and misdeeds alike, the concept of being part of some "larger world" feels far more credible in a sandbox style game. You have to develop your own motives, typically; fate (or the Game Master) isn't taking you by the hand and leading you to the waters of the story.
Things will, or will not transpire with and without you as a player.
I tend toward this approach because characters are forced to evolve and involve themselves, and many do with time. They might wander here and there, lost for a while, but they'll come to find something that motivates them, if they lacked it initially (as many oft do). Sometimes by the time they do, things are too late and the darkness has encroached - the fight is more uphill than ever, but such is the nature of reality. One's actions, or inaction, can come with great opportunity cost and detriment.
In many ways, a sandbox story is much more akin to reality... which is what I try to bend my settings to; to operate in their own realities, even if that perception of reality isn't a direct parallel to our own; of course a fantasy world's "reality" would differ from ours, as example.
While the players may or may not be the heroes, the rest of the world will keep going on with or without them. It is just they have a greater opportunity than most to shape it. That's the admirable attraction to it, or so I believe; you, by virtue of your deeds or lack there of, can be the crux of a major plot that wasn't solely focused on you.