I was watching a Chinese movie about the Monkey King (孙悟空）, and after he sprung out of the rock, he said "我出世了". I looked it up and understood it to mean "I was born". Are the two terms interchangeable or is one more formal than the other?
出世 has more use scenarios than
出生. When used for people, it's same as
出生 is used more commonly for this case); but it could be also used for things which come into being.
(1) [vouchsafe;come into being;be born]∶人的出生
(2) [come out;be published]∶问世;产生
(3) [be above worldly considerations;be not involved in the material world]∶超脱超脱人世束缚。佛教用语,佛教徒以人世为俗世,故称脱离人世束缚为出世
(4) [rise above this world]∶高出人世
出生 is used for humans and animals mostly. 出世 is used in ancient novels for heroes and great persons, also used for lifeless things like treasures and rarities.
The quick answer is that 出生 means "to be born," that is to emerge from a womb and live on. 出世 means "to be reborn," in the Hindu/Buddhist sense.