Should in the modern sense means "ought" or that you would do something, but are unable to. As in, "I should clean my room, but I'm tired." The archaic sense, as in owing an obligation, has mostly been replaced with the singular simple present: shall.
The phrase "supposed to," when paired with an infinitive suggests an obligation or an intended purpose. Such as, "I'm supposed to clean my room," or "A clean room is supposed to look nice."
So, in modern usage, "supposed to" suggests owing an obligation as opposed to "should" which, in itself, has become a suggestion.