graybeard vs old-timer

graybeard and old-timer both are nouns.

Word	Noun	Adjective	Verb	Adverb
graybeard	Yes	No	No	No
old-timer	Yes	No	No	No

As nouns, old-timer is a hyponym of graybeard; that is, old-timer is a word with a more specific, narrower meaning than graybeard:

Other hyponyms of graybeard include codger, old codger, antique, gaffer, old geezer, oldtimer, patriarch.

graybeard (noun)	old-timer (noun)
a man who is very old	an elderly man
	an experienced person who has been through many battles; someone who has given long service