childbirth vs birthing

childbirth and birthing both are nouns.

Word	Noun	Adjective	Verb	Adverb
childbirth	Yes	No	No	No
birthing	Yes	No	No	No

As nouns, birthing is a hypernym of childbirth; that is, birthing is a word with a broader meaning than childbirth:

childbirth: the parturition process in human beings; having a baby; the process of giving birth to a child
birthing: the process of giving birth

Other hypernyms of childbirth include birth, giving birth, parturition.

childbirth (noun)	birthing (noun)
the parturition process in human beings; having a baby; the process of giving birth to a child	the process of giving birth