cutwork vs embroidery

cutwork and embroidery both are nouns.

Word	Noun	Adjective	Verb	Adverb
cutwork	Yes	No	No	No
embroidery	Yes	No	No	No

As nouns, embroidery is a hypernym of cutwork; that is, embroidery is a word with a broader meaning than cutwork:

cutwork: embroidery in which the design is outlined in a buttonhole stitch and the intervening material is cut away
embroidery: decorative needlework

Other hypernyms of cutwork include fancywork.

cutwork (noun)	embroidery (noun)
embroidery in which the design is outlined in a buttonhole stitch and the intervening material is cut away	decorative needlework
	elaboration of an interpretation by the use of decorative (sometimes fictitious) detail