sick headache vs hemicrania

sick headache and hemicrania both are nouns.

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

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

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

Other synonyms of sick headache include megrim, migraine.

sick headache (noun)	hemicrania (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