edible asparagus vs herb

edible asparagus and herb both are nouns.

Word	Noun	Adjective	Verb	Adverb
edible asparagus	Yes	No	No	No
herb	Yes	No	No	No

As nouns, herb is a hypernym of edible asparagus; that is, herb is a word with a broader meaning than edible asparagus:

edible asparagus: plant whose succulent young shoots are cooked and eaten as a vegetable
herb: a plant lacking a permanent woody stem; many are flowering garden plants or potherbs; some having medicinal properties; some are pests

Other hypernyms of edible asparagus include herbaceous plant.

edible asparagus (noun)	herb (noun)
plant whose succulent young shoots are cooked and eaten as a vegetable	aromatic potherb used in cookery for its savory qualities
	a plant lacking a permanent woody stem; many are flowering garden plants or potherbs; some having medicinal properties; some are pests