breakthrough vs brainwave

breakthrough and brainwave both are nouns.

Word	Noun	Adjective	Verb	Adverb
breakthrough	Yes	No	No	No
brainwave	Yes	No	No	No

As nouns, brainwave is a hypernym of breakthrough; that is, brainwave is a word with a broader meaning than breakthrough:

breakthrough: a productive insight
brainwave: the clear (and often sudden) understanding of a complex situation

Other hypernyms of breakthrough include brainstorm, insight.

breakthrough (noun)	brainwave (noun)
a penetration of a barrier such as an enemy's defense	the clear (and often sudden) understanding of a complex situation
making an important discovery	(neurophysiology) rapid fluctuations of voltage between parts of the cerebral cortex that are detectable with an electroencephalograph
a productive insight