sympathizer vs admirer

sympathizer and admirer both are nouns.

Word	Noun	Adjective	Verb	Adverb
sympathizer	Yes	No	No	No
admirer	Yes	No	No	No

As nouns, admirer is a hypernym of sympathizer; that is, admirer is a word with a broader meaning than sympathizer:

sympathizer: someone who shares your feelings or opinions and hopes that you will be successful
admirer: a person who backs a politician or a team etc.

Other hypernyms of sympathizer include booster, champion, friend, protagonist, supporter.

sympathizer (noun)	admirer (noun)
someone who shares your feelings or opinions and hopes that you will be successful	someone who admires a young woman
a person who commiserates with someone who has had misfortune	a person who backs a politician or a team etc.
	a person who admires; someone who esteems or respects or approves