The reason it changes to the local copy is so that you can still view/edit the form when you are not connected to the network - this is a very key feature in InfoPath's functionality.
There's not much difference between saving and publishing, but I do know that publishing "finalizes" several things and updates URLs and URNs according to where you publish it, whereas saving leaves things in a "raw" state. I usually save locally when I'm designing a new template and then when everything is ready to go, publish to a shared resource.
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Matt Faus
InfoPath Dev Texas
Matt Faus / Microsoft InfoPath MVPQdabra® Software / Streamline data gathering to turn process into knowledge