enlist vs sign up

enlist and sign up both are verbs.

Word	Noun	Adjective	Verb	Adverb
enlist	No	No	Yes	No
sign up	No	No	Yes	No

As verbs, sign up is a hypernym of enlist; that is, sign up is a word with a broader meaning than enlist:

enlist: join the military
sign up: join a club, an activity, etc. with the intention to join or participate

enlist (verb)	sign up (verb)
join the military	join a club, an activity, etc. with the intention to join or participate
engage somebody to enter the army	engage by written agreement
hire for work or assistance