airmail vs mail service

airmail and mail service both are nouns.

Word	Noun	Adjective	Verb	Adverb
airmail	Yes	No	Yes	No
mail service	Yes	No	No	No

As nouns, mail service is a hypernym of airmail; that is, mail service is a word with a broader meaning than airmail:

airmail: a system of conveying mail by aircraft
mail service: the system whereby messages are transmitted via the post office

Other hypernyms of airmail include mail, post, postal service.

airmail (noun)	mail service (noun)
a system of conveying mail by aircraft	the system whereby messages are transmitted via the post office
letters and packages that are transported by aircraft

airmail (verb)	mail service (verb)
send or transport by airmail