Air pressure is the intrinsic pressure in a quantity of air. It can come from any number of sources. Perhaps there is a closed cylinder of air with a piston compressing it isothermally. Or perhaps the air in the tire in your car is under pressure due to the weight of the car and the surface tension of the tire.
Atmospheric pressure is air pressure due solely to the weight of the air above you in the atmosphere. It is typically around 105 N/m2105 N/m2 on Earth, but can vary with altitude and weather. It is just a "natural," ubiquitous special case of air pressure in physical situations taking place inside an atmosphere.
