of a sudden vs on the spur of the moment

of a sudden and on the spur of the moment both are adverbs.

Word	Noun	Adjective	Verb	Adverb
of a sudden	No	No	No	Yes
on the spur of the moment	No	No	No	Yes

As adverbs, of a sudden and on the spur of the moment are synonyms defined as:

of a sudden and on the spur of the moment: quickly and without warning
of a sudden and on the spur of the moment: happening unexpectedly
of a sudden and on the spur of the moment: on impulse; without premeditation

Other synonyms of of a sudden include abruptly, all of a sudden, dead, short, suddenly.

of a sudden (adverb)	on the spur of the moment (adverb)
quickly and without warning; happening unexpectedly; on impulse; without premeditation	quickly and without warning; happening unexpectedly; on impulse; without premeditation