wailing vs weeping

wailing and weeping both are nouns.

Word	Noun	Adjective	Verb	Adverb
wailing	Yes	Yes	No	No
weeping	Yes	Yes	No	No

As nouns, weeping is a hypernym of wailing; that is, weeping is a word with a broader meaning than wailing:

wailing: loud cries made while weeping
weeping: the process of shedding tears (usually accompanied by sobs or other inarticulate sounds)

Other hypernyms of wailing include crying, tears.

wailing (noun)	weeping (noun)
loud cries made while weeping	the process of shedding tears (usually accompanied by sobs or other inarticulate sounds)

wailing (adjective)	weeping (adjective)
vocally expressing grief or sorrow or resembling such expression	showing sorrow
	having branches or flower heads that bend downward