These two phrases are related, however 'to be supposed to' implies that another person expects or demands that you do something. For example: 'I don't have any food left in the refrigerator...I should really go to the supermarket!' (In this case, it means 'I really need to...') 'I'm supposed to be at work at 8am tomorrow, but I can't even get out of bed today. I'm going to call in sick.' (In this case, your boss expects you to be at work)

Supposed to = obligation. I am supposed to complete my English assignment. Should = not obliged, but may benefit from. If you are unsure of how to spell a word, you should check a dictionary. Make sense?