exhaustion vs brain-fag

exhaustion and brain-fag both are nouns.

Word	Noun	Adjective	Verb	Adverb
exhaustion	Yes	No	No	No
brain-fag	Yes	No	No	No

As nouns, brain-fag is a hyponym of exhaustion; that is, brain-fag is a word with a more specific, narrower meaning than exhaustion:

Other hyponyms of exhaustion include inanition, frazzle, mental exhaustion.

exhaustion (noun)	brain-fag (noun)
the act of exhausting something entirely	exhaustion that affects mental keenness
serious weakening and loss of energy
extreme fatigue