One thing is you can do is look around at other homes that have had concrete and asphalt for driveways for a few years where you live. Here, in PA, the vast majority of driveways are asphalt for a couple of reasons. They settle better due to the extremes in temperature over the course of the year. They're easier to repair, in general. Usually, in climates that are 'warm' year-round, concrete is used because more often than not the weather doesn't undergo the extremes in temperature changes found in, say, the NorthEast. And in warmer climates, concrete is not as hot to walk on in summer as asphalt.
You might want to keep in mind that the condition and age of your 'neighbors' driveways should help in this decision. And one last tip ... I suggest that if you haven't yet had your shed delivered, or anything else delivered that requires a heavy-duty truck using your driveway, I wouldn't do a thing to your current driveway until everything is delivered/installed. Cement trucks, 28' flat bed deliver trucks, furniture trucks, etc., etc., will put a lot of weight on your driveway and possibly damage it, concrete or asphalt. They aren't responsible either for the damage, and wouldn't be if you asked them to put it in writing. (They'll tell you to meet them at the street if you don't want to take a chance of them damaging your new driveway!)