They’re very similar! I would say that the difference is that “What should I do” asks for a suggestion, while “What am I supposed to do” usually refers to rules that need to be followed

“What should I do?” is usually a request for advise as to the best course of action. “What am I supposed to do?” is literally asking what is expected of me (by others, by the rules, etc.), and is often used rhetorically to mean the speaker doesn’t see any good options.

They have subtle differences, although they are often used interchangeably.. “Should” is used often as an advice or a suggestion. “Supposed to” is used to indicate that it is expected out of you. "What should I do?" You are probably asking for advice. While "What am I supposed to do?" You might be asking what is excepted of you or what ate you required to do.