From an engineering perspective, concrete is modeled as a rigid pavement and asphalt is a flexible pavement.
Concrete is harder and more brittle than asphalt. Concrete generally cannot tolerate significant ground movement without cracking. The addition of steel reinforcing to concrete can help to mitigate concrete's brittleness to some extent, but Jan is correct in saying that once concrete has cracked or spalled, it cannot be easily repaired. Generally an entire panel must be completely replaced.
Asphalt is less expensive than concrete to install initially but the surface will degrade over time due to the evaporation of volatile aromatic hydrocarbons so must be maintained on a regular schedule whereas a concrete surface requires no regular maintenance. Asphalt is a weaker material than concrete but can be made to support equal loads by constructing a thicker pavement. Asphalt will soften under high temperatures so warm climates can be a problem for asphalt. Because asphalt is flexible it is a better choice over soils that are prone to movement or over soils that would be too expensive to treat to raise their bearing capacity.
Concrete reflects light better than asphalt so will have better visibility at night. For highways, asphalt is a quieter surface than concrete although that would not be a consideration for aviation purposes.
The choice of whether to use concrete or asphalt must consider all these variables of initial cost vs. long term maintenance, soil condition, visibility, climate, local budgetary considerations, local availability of materials, etc. Each installation will be different.
I have supervised drilling cores from the runway and taxiways at San Diego International Airport (KSAN). It has alternating layers of concrete and asphalt in some locations although the top surface is asphalt. The total thickness varies but is generally at least 18" thick. The airport sits on the former outlet to the San Diego River and is only a short distance from the bay, so the sub soils are weak alluvial silts that are saturated a short distance below grade. Yet the taxiways are mostly concrete. Go figure.