As a non-native speaker, I should believe that there is a slight difference between these two sentences. I shall provide you with an explanation of the meanings of each sentence (with regard to what I believe they mean!):
1) Firstly, you should know that ''be bound to (do something)'' is an idiom used to indicate that a person is likely to or is certain/obliged to do something. So, in the sentence ''With gas prices going up, you are bound to save money in the end'', the idea conveyed is that of an obligation or likeliness regarding what a person has to do, given the actual facts.
2) ''Must be (doing something)'' = You use ''must'' with ''be'' and an ''-ing form'' to say that you believe something is happening, for example: ''He isn't in his office. He must be working at home'' or ''You must be getting tired''. So, in the second statement you provided: ''With gas prices going up, you must be saving money in the end'', the idea conveyed is that of a belief or certainty about a situation = in this case, you believe that the person is really saving money so as to be able to deal with the actual problem (the gas rising rates).
Hope that helps! :)