KidSnide
Adventurer
Nah.
Game elements come first for both. You know....settings, NPCs, the plots of NPCs, etc. The story-oriented GM then determines what should happen with those plot elements ("What would make a cool/dramatic scene is if the PCs.....") whereas the sandbox-oriented GM then says "I wonder what the PCs will do when presented with these elements?" and crafts some more elements.
Well, speaking as a story-oriented hybrid GM, game elements are a big part of pre-planning, but sometimes I pick moments and try to build the game to them.
For example, about seven years into my last campaign, the PC party was hunting for the great individual they thought to apotheosize (as it were) into an allied demi-god (Hope) if put into the right situation. When she went missing they went after her, to find her dead -- an impossible event according to their prophesy. At that point, they realized the prophecy was wrong and it was *the PCs* who could (collectively) apotheosize into the a demi-god (in their case, Unity).
Sure, I guess you could see this as a game element, but it was really the moment I was driving to -- that single point when the players realized that they weren't finding someone who could save the world. They had to save the world themselves. (Then, everyone gained 3 levels and had a massive set fight against a hated enemy they used to think was unbeatable.)
It was a miracle that they players actually figured it out at the same time (and while I had them standing in a circle, too). But, in my book at least, this was an example of wanting a dramatic effect and then building plot elements (and manipulating the hell out of the players) to get the moment. As a GM, you need to sparing with this sort of idea. (I think of it as a once every 1-3 years kind of ploy.) But it's definitely possible.
-KS