Both offer jet massages while bathing, but whirlpools mainly use jetted water. Air baths feature jets that shoot air streams.
Whirlpools have fewer jets but provide a more intense sensation, whereas air baths have numerous smaller jets throughout that create a gentler, bubblier feeling (source).
Many people prefer the more intense therapeutic feeling that whirlpools offer, but air baths have their own merits. Some modern whirlpools use a combination of both air and water, so you get the best of both worlds. This is largely a matter of personal preference.