Well, polish can be a verb or a noun. The noun refers to the stuff in the little tins that we put on our shoes. We then polish (v) them to make them shine. Or we shine them to make them Polish.
Wait, that's not right. The verbs, polish and shine, are, in my experience, interchangeable. There's not some thing we have done to our shoes (or do ourselves) that is a polish rather than a shine. That's what I took the OP to be asking. My answer is that, as verbs, they mean the same thing.