I've been thinking about the sandbox approach to setting/adventure design as compared to the approach, and here are my thoughts so far:
Characters in a Sandbox
The classic example of this is the D&D trope of low-level characters running through a gauntlet of rooms and adventures, with those who manage to survive and level up eventually getting a chance to build up relationships with other PCs and recurring NPCs -- perhaps even allowing their backstories to intrude on their current adventures. It may be that these NPCs eventually spark some adventures, but the PCs themselves -- ideally -- should be the prime motivators of the next set of adventures.
There seem to be two approaches to handling adventures: (1) at the end of the next session, agree what the next adventuring site will be so that the DM can prepare; or (2) allow the DM to read as much of the setting and adventure locations as possible, and trust to the random tables and improv abilities of the DM to fill in the gaps. These approaches are not mutually exclusive, of course. The DM can employ one or the other or a combination of both as long as he and the players continue to have fun.
Is there a story in this? There can be, if the DM prefers. The PCs aren't necessarily required to be involved in it though -- and that storyline will tend to continue if the PCs don't meddle in it. Examples: increasingly powerful goblin raiders, more and more shadowy creatures roaming the forests, strife between the nobles of the land over a matter of succession, and so on. The players can occasionally listen to town criers and pick up local scuttlebutt at the taverns, but -- like true wandering adventurer archetypes -- only get involved when they feel like it.
In a way, from the point of view of the movers and shakers of the realm (good, evil, or in-between) the adventurers are the wildcards. They are seen as tools, pawns, and even scapegoats if they even register in the minds of the powerful forces of the land.
Once they become notable, however, unless they perform a disappearing act or manage to hide their identities, then the wise movers and shakers will likely to try to determine where these ruffians' loyalties lie. This would be a natural transition point to a story-based campaign.
Ultimately, there is a sense of empowerment here -- players can bug out of a dungeon at any time (if they carefully secured a way out) and move on to some other less dangerous challenge.
Characters in a Story
Just as the most negative portrayal of sandbox play is a run-of-the-mill Computer RPG with stock characters who don't remember who you are unless they're programmed to and unchanging, re-spawning monsters from given locations, the most negative portrayal of storyline play is one where the PCs are expected to perform their roles exactly as the adventure / story crafter expects them to. There is an illusion of player decision, when in fact there is none (or very little).
Good story campaigns should take into consideration the various decisions of the PCs and extrapolate the reactions of the NPCs accordingly. The PCs may not have an impact on the world at large, but on their own stories their efforts and successes should matter. For example, if a player comes up with a clever way to kill a major NPC villain early in the storyline, it should stand. The NPC(s) who step into the vacuum should not be mere carbon copies of the deceased either, and should have their own foibles and personas that make the resolution different (perhaps easier given some PC strategems, perhaps harder). Ultimately that triumph of the players should in the end make a palpable difference.in the rest of their storyline.
In fact, a very rigid storyline adventure comes across like a linear dungeon with only one possible entrance and one possible exit from each room that PCs must trudge on inexorably through, hoping to roll well enough (since their decisions -- tactical or strategic -- make no real difference) to get through to the end.
PC empowerment is trickier, though not impossible in story-based adventures in campaigns. The difficulty may lie in the fact that NPCs have just as many options as the PCs, and this can overwhelm the DM, who not only has to roll up these NPCs, but roleplay each and everyone of them as well (stupid or smart, vindictive or forgiving, generous or greedy), and know their pasts and families and opponents should the PC suddenly engage them in conversation.