1. -아/어지다 - become some quality; turn certain way. Used with an adjective (밝아지다, 추워지다, etc). 2. -게 되다 - come to a certain situation or state. Used more commonly with a verb (이 회사에서 일하게 되다. 일이 어렵게 되다). (1) describes something changing state at the very moment, as in 날이 어두워진다, 얼굴이 붉어졌다. It usually involves a natural, predictable phenomenon. (2) usually implies an event of going through a process to finally end up in certain situation. For example, 오늘부터 여려분들을 가르치게 되었어요, 집세를 내지 못해 쫓겨나게 되었다, 이번에 새로 사업을 하게 됐어요 all describe the current situation the speaker finds themselves in following an unmentioned endeavor or happenings that led to it.

The first is used for adjectives. To be come pretty 예뻐지다, to become tired 피곤해지다 etc. The second one I think maybe is used for nouns, but I'm not sure.