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.

I agree with some others: "to be meant to" sounds like fate/destiny. It's something you are happy to do. (Unless you use it in the negative, but it still refers to fate/destiny) "To be suppose to" sounds like something somebody wants you to do (like a task, assignment, or chore). You're probably not that excited to do what you're suppose to. So the meaning is probably about the same but the nuance/mood of each is different (like the wife example portrays). I would say that we don't use "to be meant to" very much though. However we use just plain old "meant to V" a lot! "I meant to give you this sooner" "Meant to" means "intended to" and implies you forgot or didn't get the chance to. "I intended to give you this sooner but I forgot".

I am supposed to = sounds like you have to do a chore. I am meant to = sounds a bit more poetic, like it's your destiny.

I think the only different of these two words would only occur when you saying this to your wife, "I am meant to marry you" and "I am suppose to marry you". You wife will love you if you using the first sentence and kill you using the second.