To be honest, this is something I've been wondering about as well. I'm thinking about joining a sandbox game, but I'm not sure what to expect. A few years back I was in a game that I think might have been a city-based sandbox, but it was extremely unsatisfying. We each crafted intricate back stories, but then spent each session wandering around the town doing random odd missions for a local barkeep without any idea why we were doing them. There was no reason behind what we were doing, other than he was our patron in the town. When we tried pursuing parts of our backstory, we just kept running into dead ends with no leads to follow. In case you can't tell, I wasn't happy with the game and ended up quitting.
When I run a campaign, I typically present several clear different avenues for the PC's to explore, but when they've chosen one I like to create a story behind their choice. So the players have a lot of control over where the story goes, but I actively work to make sure their choices create an engaging story. Is that typical of what people think of when they think of a "sandbox" game? I don't really know. I certainly guide the players by creating plots and presenting plot hooks in the way of the players, but I also try not to force the players to go any certain way, and I'm prepared to improvise if they want to go in a different direction. It's just been my experience that they rarely do! Even when the players are following a plot line, I try to create situations where the players have to choose between several different exclusive paths. So I guess you could call my campaign style a "choose-your-own-adventure" style. Again, I don't know if that is typical of sandbox play.
So I'll join in with the OP and ask, for those who spend a lot of time in a sandbox campaign, what are the distinguishing characteristics that identify it as a sandbox? If someone sees a sandbox game advertised and wants to join, what should they expect?