backache vs lumbago

backache and lumbago both are nouns.

Word	Noun	Adjective	Verb	Adverb
backache	Yes	No	No	No
lumbago	Yes	No	No	No

As nouns, lumbago is a hyponym of backache; that is, lumbago is a word with a more specific, narrower meaning than backache:

backache: an ache localized in the back
lumbago: backache affecting the lumbar region or lower back; can be caused by muscle strain or arthritis or vascular insufficiency or a ruptured intervertebral disc

Other hyponyms of backache include lumbar pain.

backache (noun)	lumbago (noun)
an ache localized in the back	backache affecting the lumbar region or lower back; can be caused by muscle strain or arthritis or vascular insufficiency or a ruptured intervertebral disc