The recent Adventurers vs Heroes post was just a random musing, but I've actually got more to say on the matter.
It's not just about the characters - the adventurer and the hero also inhabit different worlds. In this column by Chris Perkins (I think his use of the word railroad is problematic, but try to look past the words), he shows how the heroes need signposts to direct them towards their role in the world.
For a hero to be a hero, she needs to stand up where no one else would (or can). The hero has to stand up in the face of rising darkness. For the hero to exist, the world needs a morally unambiguous evil. It needs this one bad thing (or two or three, but no more) that's threatening everything, so there's no question about it - so that the hero can align herself in opposition and towards this wrong. It is saying (to the character and the player): here's where the game is. It's what the 4E says. It's what Storming says. The world (aka The GM) erects signposts saying "play in this direction".
In The Dying Earth however, nobody can stop the dying sun from going out. It's going to be eternal darkness soon, and that's it. So the adventurers pick whatever direction based on circumstance, their petty ambitions or wanderlust. The signposts don't point in one direction but in many different directions and every time the adventurers pick a direction, the signposts change, disappear, new ones grow. There are no unambiguous large threats, and if there is one, the adventurers are probably picking up their stuff and leaving by the back door.
The hero inhabits the world of Paths, Unambiguous Threats/Villains. The adventurer inhabits a world of Locations, Conflicting Interests. In other words: the former are arc-based, the later are sandboxes.
Although both can trace their roots to forms of fiction (picaresque vs epic tale etc.) I think they're also a very practical response to the realities of play, specifically the GM's work. To prepare a whole world in which the characters live is pure madness on top of being basically impossible. The heroic game solves this by allowing the GM to prep that one thing (or at least a narrow selection of things), focusing the direction of play. The adventurer game drops the narrative arc in favor of freedom - it offers a whole world, but one that is random and procedurally generated. (People used to running heroic-prep games often don't understand how an adventuring-sandbox is even possible.)
In the case of D&D (which is what I'm using as the baseline for this talk), both approaches still fulfill the same agenda, but from different angles. Perhaps more on that a different time.