emblem vs crest

emblem and crest both are nouns.

Word	Noun	Adjective	Verb	Adverb
emblem	Yes	No	No	No
crest	Yes	No	Yes	No

As nouns, crest is a hyponym of emblem; that is, crest is a word with a more specific, narrower meaning than emblem:

emblem: special design or visual object representing a quality, type, group, etc.
crest: (heraldry) in medieval times, an emblem used to decorate a helmet

Other hyponyms of emblem include colophon, colors, colours, Cross, device, flag, heraldry, skull and crossbones, totem, totem pole.

emblem (noun)	crest (noun)
special design or visual object representing a quality, type, group, etc.	a showy growth of e.g. feathers or skin on the head of a bird or other animal
a visible symbol representing an abstract idea	(heraldry) in medieval times, an emblem used to decorate a helmet
	the center of a cambered road
	the top line of a hill, mountain, or wave
	the top or extreme point of something (usually a mountain or hill)

emblem (verb)	crest (verb)
	reach a high point
	lie at the top of