I can see where Yang and Sara are coming from with the chore v. destiny distinction, but, personally, I don't think that there is any difference between the way we use these expressions in everyday modern English.
If you have a problem, and you don't know how you are expected to behave, you can say either 'So what am I supposed to do now?' or 'So what am I meant to do now?', and the meaning is the same. I wouldn't worry about the difference, if I were you. You can use the two expressions in exactly the same way.