Differences between Slope of budget line and MRS?
Not really, you're right in that (loosely speaking) the MRS is the amount of one good someone is willing to give up in order to get an additional unit of another good. However, the slope of the budget line measures the amount of one good someone has to give up in order to get an additional unit of another good. 

In the first case, only preferences matter, whereas in the second only prices are relevant.