wrap vs do up

wrap is a noun but do up is not a noun.

Word	Noun	Adjective	Verb	Adverb
wrap	Yes	No	Yes	No
do up	No	No	Yes	No

As verbs, do up is a hyponym of wrap; that is, do up is a word with a more specific, narrower meaning than wrap:

Other hyponyms of wrap include parcel, cere, shrinkwrap, gift-wrap, shroud.

wrap (noun)	do up (noun)
cloak that is folded or wrapped around a person
the covering (usually paper or cellophane) in which something is wrapped
a sandwich in which the filling is rolled up in a soft tortilla

wrap (verb)	do up (verb)
arrange or fold as a cover or protection	use special care in dressing, making-up, etc.
enclose or enfold completely with or as if with a covering	wrap for decorative purposes
arrange or coil around
crash into so as to coil around