区别是：
be supposed to 表示“理应”强调客观会做，“猜想会”，常用于被动语态中。
should 表示应该，强调主观上应该做。
例句：
be supposed to
1、You're supposed to be my friend!
你还是我朋友呢！
2、We're supposed to be running a business here. I've got work to do.
我们在这里应该是要经营公司。我有一堆的事情要做。
3、Don't try to tell me what I'm supposed to be feeling.
别跟我说我到底该作何感想。
4、The performance seemed to me unpardonable, a contradiction of all that the Olympics is supposed to be.
在我看来，这次的表现是不可宽恕的，和奥运精神完全相悖。
should
1、I should exercise more.
我应该多锻炼一些。
2、All visitors should register with the British Embassy.
所有游客都必须到英国大使馆登记。
3、I should have gone this morning but I was feeling a bit ill.
今天早晨我本应该走的，但是我觉得有点不舒服。
4、You should have no problem with reading this language.
阅读这种语言你应该没问题。
