analgesic vs medicinal drug

analgesic and medicinal drug both are nouns.

Word	Noun	Adjective	Verb	Adverb
analgesic	Yes	Yes	No	No
medicinal drug	Yes	No	No	No

As nouns, medicinal drug is a hypernym of analgesic; that is, medicinal drug is a word with a broader meaning than analgesic:

analgesic: a medicine used to relieve pain
medicinal drug: (medicine) something that treats or prevents or alleviates the symptoms of disease

Other hypernyms of analgesic include medicament, medication, medicine.

analgesic (noun)	medicinal drug (noun)
a medicine used to relieve pain	(medicine) something that treats or prevents or alleviates the symptoms of disease

analgesic (adjective)	medicinal drug (adjective)
capable of relieving pain