In practice, you can use these interchangeably. But there is a very subtle difference between 1: "What am I supposed to do?" and 2: "What should I do?". Sentence 1 implies more a set of external rules that are 'supposed' to be followed. Sentence 2 is not quite as narrow and could refer to somebody making a freer choice, or being guided by their own conscience.
For example. Imagine you must decide whether to buy an apple or an orange. If you say: "I should buy the apple", this implies that the apple is the best choice for reasons that you have thought about yourself; maybe you prefer the taste and the orange is slightly rotten.
If, on the other hand, you say: "I am supposed to buy the apple", this more strongly implies that some predetermined destiny is guiding you to buy the apple.
Another example: If you were baking a cake with a friend and wanted to ask them about the next step in the recipe, if you say: "What am I supposed to do next?" It is clear that you are asking about the next step in the recipe.
However, if you ask "What should I do next?" there is a chance that you are not only talking about the recipe and may be asking your friend what they think you should do next in a more general, abstract sense.
In everyday speaking, should is usually a safer bet, but both are fine and understandable in any situation.
Hope that isn't too confusing, like I said the distinction is very subtle and most people wouldn't even notice you use them interchangeably :)