We use “should” to make a suggestion, to say something is a good idea. We use “supposed to” to indicate that something is almost an obligation. If it were really an obligation (with consequences for not doing it), we would say “have to,” “need to,” or “must.” (Note that “must” is usually used for speculation, meaning the speaker is almost sure of something.)

'Should' is used to make a suggestion, that is, to express that something seems to be a good idea. As for ''to be supposed to'', its usage conveys the idea that something is pretty much an obligation, or some kind of duty. Examples; 1) ''You should really try on that dress! It fits you perfectly!'' 2) ''I was supposed to pick her up at 19:00, but I just got stuck in traffic!'' 3) ''We are not supposed to walk into the woods by ourselves! Mom told us as much!'' 4) ''She should be happy! - she just got promoted!'' 5) ''It's so cold out there! I should take a coat along with me - the weather is not likely to change today!'' Hope that helps!