culinary art vs cookery

culinary art and cookery both are nouns.

Word	Noun	Adjective	Verb	Adverb
culinary art	Yes	No	No	No
cookery	Yes	No	No	No

As nouns, cookery is a hypernym of culinary art; that is, cookery is a word with a broader meaning than culinary art:

culinary art: the practice or manner of preparing food or the food so prepared
cookery: the act of preparing something (as food) by the application of heat

Other hypernyms of culinary art include cooking, preparation.

culinary art (noun)	cookery (noun)
the practice or manner of preparing food or the food so prepared	the act of preparing something (as food) by the application of heat