rally vs pull together

rally is a noun but pull together is not a noun.

Word	Noun	Adjective	Verb	Adverb
rally	Yes	No	Yes	No
pull together	No	No	Yes	No

As verbs, pull together is a hypernym of rally; that is, pull together is a word with a broader meaning than rally:

Other hypernyms of rally include collect, garner, gather.

rally (noun)	pull together (noun)
(sports) an unbroken sequence of several successive strokes
the feat of mustering strength for a renewed effort
an automobile race run over public roads
a large gathering of people intended to arouse enthusiasm
a marked recovery of strength or spirits during an illness

rally (verb)	pull together (verb)
return to a former condition	assemble or get together
harass with persistent criticism or carping
call to arms; of military personnel
gather
gather or bring together