drinking vessel vs mug

drinking vessel and mug both are nouns.

Word	Noun	Adjective	Verb	Adverb
drinking vessel	Yes	No	No	No
mug	Yes	No	Yes	No

As nouns, mug is a hyponym of drinking vessel; that is, mug is a word with a more specific, narrower meaning than drinking vessel:

Other hyponyms of drinking vessel include loving cup, stoup, tankard.

drinking vessel (noun)	mug (noun)
a vessel intended for drinking	with handle and usually cylindrical
	the human face (`kisser' and `smiler' and `mug' are informal terms for `face' and `phiz' is British)
	a person who is gullible and easy to take advantage of
	the quantity that can be held in a mug

drinking vessel (verb)	mug (verb)
	To stare awkwardly
	rob at gunpoint or with the threat of violence