obstacle vs hang-up

obstacle and hang-up both are nouns.

Word	Noun	Adjective	Verb	Adverb
obstacle	Yes	No	No	No
hang-up	Yes	No	No	No

As nouns, hang-up is a hyponym of obstacle; that is, hang-up is a word with a more specific, narrower meaning than obstacle:

obstacle: something immaterial that stands in the way and must be circumvented or surmounted
hang-up: an unforeseen obstacle

Other hyponyms of obstacle include barrier, roadblock, hitch, rub, snag, hurdle, stymie, stymy.

obstacle (noun)	hang-up (noun)
an obstruction that stands in the way (and must be removed or surmounted or circumvented)	an unforeseen obstacle
something immaterial that stands in the way and must be circumvented or surmounted	an emotional preoccupation