Asphalt is a "soft" material that bends and flexes under the heavy load of a jumbo jet. Concrete will simply crack and break if enough pressure is placed on it.
Concrete withstands heat much better than asphalt, which can actually melt on a really hot day or from the heat of a jet's turbine exhaust.
Porous asphalt absorbs water better than concrete, which is susceptible to potentially dangerous standing water.
Under the right conditions, the rough surface of asphalt can cause jet tires to burst. Concrete's smoother surface makes it harder to stop a jet, however.
Some airports use a combination of the two surfaces: concrete for taxiways and the gates where jets will be parked for long periods and asphalt for the actual runway.
