I think they are interchangeable in most situations. " Supposed to" might sound a bit colloquial, but it's commonly used. Let's make a deal, we'll use "should" in formal writing and either one informally.

Just to add to Julie's comment, one thing I would say about 'supposed to' is that, to me, it implies that either you don't want to do the task, or you are considering not doing it.

I agree with the earlier poster, these are fairly equivalent and interchangeable in most circumstances. There is some nuanced distinction. "Supposed to" invokes an unseen authority, while "should" is more universal or pertains to general knowledge. For instance, a child is not supposed to watch TV until chores are done, while everyone knows you shouldn't eat fast food if you want to stay thin. Moreover, "supposed to" can be used by adults to appear cute or suggest something childish or playful. For instance, my girlfriend once told me, when we were first getting acquainted, that she was "not supposed to use Carmex or any other medicated lip balm" because she uses too much of it. By saying "not supposed to" instead of "shouldn't," she was being cute and characterizing it as "bad," like a child, while also invoking past social interactions and a running joke with her friends who told her not to use it. Finally, "should" especially is passive voice, so you might consider rewording. Ask yourself, who "shoulds" you in this circumstance? Is there a more direct way of saying it?

"Supposed to" and "should" are very close in meaning and could be considered interchangeable in most situations. Both imply some expectation or obligation whether it be a rule, law, custom, tradition, or just common courtesy. However, one difference is that "should" can also be used when someone is simply expressing their opinion. "I think you should go." In this case there is no expectation or obligation, simply someone's opinion about what you should do.

'suppose' expresses more doubt in the sentence while 'should' expresses a conditional sense.