old age vs dotage

old age and dotage both are nouns.

Word	Noun	Adjective	Verb	Adverb
old age	Yes	No	No	No
dotage	Yes	No	No	No

As nouns, dotage is a hyponym of old age; that is, dotage is a word with a more specific, narrower meaning than old age:

old age: a late time of life
dotage: mental infirmity as a consequence of old age; sometimes shown by foolish infatuations

Other hyponyms of old age include second childhood, senility, autumn years, decrepitude.

old age (noun)	dotage (noun)
a late time of life	mental infirmity as a consequence of old age; sometimes shown by foolish infatuations