What is the difference between 부서지다 and 망가지다 ?Feel free to just provide example sentences.

부서지다 is usually used when something is cracked or split into pieces or in a bad condition, and 망가지다 is used when something or someone is in a condition of unableness to perform(But it is not that frequently used to explain human condition). Yet, 부서지다 can be used as a term of 망가지다 when something cannot be used or perform its work because it is 부서짐. But the usages of these words are not restricted, they can be used in many different ways.
To clarify, I'll explain with sample sentences.
부서지다:
-문고리가 부서져서 문을 열 수 없다. (The doorknob fell out, so I can't open the door.)
-얼음조각을 망치로 내려쳤더니 부서졌다. (Hitting it with a hammer, the ice block broke into pieces)

망가지다:
-이 연장은 망가져서 더 이상 쓰일 수 없다. (This tool is on the blink and cannot be used further.)
-그녀의 몸은 오랜 세월의 암 투병 끝에 망가지고 말았다. (Suffering through a long time of fighting against cancer, her body system was entirely torn down.)

But you must know they are used in very various circumstances and subjects.