sick headache vs migraine

sick headache and migraine both are nouns.

Word	Noun	Adjective	Verb	Adverb
sick headache	Yes	No	No	No
migraine	Yes	No	No	No

As nouns, sick headache and migraine are synonyms defined as:

sick headache and migraine: a severe recurring vascular headache; occurs more frequently in women than men

Other synonyms of sick headache include hemicrania, megrim.

sick headache (noun)	migraine (noun)
a severe recurring vascular headache; occurs more frequently in women than men	a severe recurring vascular headache; occurs more frequently in women than men
a headache accompanied by nausea