Frankfurt Airport; a mosaic of concrete and asphalt
The runways at Frankfurt Airport are asphalt (old satellite imagery shows that they were concrete before), and the airport handles plenty of Superjumbos, the Airbus A380, so it's not a matter of handling heavy planes. All runways in big airports, even those covered in asphalt, have reinforced concrete foundations that vary in depth. They're deepest where the touchdown zones are located, shallower elsewhere. That's why runway extensions usually just add extra takeoff distance, but the landing zones remain the same, creating a displaced threshold.
Displaced threshold; can exist for other reasons
You'll see almost all the parking spots and holding areas for the runways are concrete, as well as the runway exits. This is where a plane is expected to be stationary. Concrete handles such stationary loads pretty well, while on a warm day the asphalt would deform.
Asphalt is called a flexible surface; concrete is called a rigid surface. Pavement classification number (PCN) is a standard used in combination with the aircraft classification number (ACN) to indicate the strength of a runway, taxiway, or apron (ramp).
Exposed concrete is notorious for cracking when exposed to extreme high and low temperatures. It's also more expensive to install and repair. But overall can be cheaper to install if you're already laying down concrete elsewhere, like most new airports, they just go for concrete everywhere, but it depends on where that new airport is located—
Hot Dubai; asphalt almost everywhere except parking areas and hold short areas
LAX; good climate and budget; concrete almost everywhere a plane touches
The variables come down to climate, use, load, budget, job creation, maintenance, etc.
A single runway (like at PBG) can also have both surfaces
Airport diagram showing the PCN for all runways; Flexible and Rigid
(NASA) Another point in favor of an asphalt runway, is that it's more gentle on the tires during touchdown (since it's flexible).
You can find runway data at airnav.com for US airports, elsewhere you'll need to check the airport's AIP, like this one for Amsterdam; check the section 2.12 RUNWAY PHYSICAL CHARACTERISTICS. Or try searching for a nonofficial runway data repository, there could be one.