Each has its own advantages and disadvantages. Asphalt is typically easier to fix as you can fix a very small area should you have a problem. Ice will melt on asphalt quicker than it will on concrete. Asphalt is typically less expensive up front, however sealer must be applied every 4-5 years which in the long run may make the costs equal.
Concrete is stronger but at some point most likely will crack. When fixing these areas you are typically replacing a joint to joint area which ends up being a larger repair to make everything look symmetrical. Concrete does have no maintenance for its life with the exception of periodically pressure washing the driveway to keep it clean.