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An air pump is a device which is use for pushing air. Whereas an air compressor is a device that work as an converter, as it converts power into potential energy which is stored in pressurized air.
However, Pumps and compressors share very similar mechanisms, and does perform the same action, but the dissimilarity comes with respect to different fluid regimes. Compressors operate on compressible fluids, typically gases. On the other hand, pumps operate on fluids, typically liquids. Compressors develop a very high pressure rise against a closed system whereas; pumps are designed to develop relatively little pressure against a free-flowing system. They are often used in continuous-flow operation, while many lower-end compressors must have intermittent duty cycles. Moreover, to shut off the compressor when desired pressure is obtained, it has got feedback sensor, while pumps have a fixed design and operate freely across their performance curve as conditions change.
Howbeit, from jackhammers to air conditioners, all kinds of machines use pumps and compressors to move liquids and gases from place to place, just like in vehicles, they both are used.