Hi - yes there is a SharePoint submit functionality in InfoPath. Are you using InfoPath 2007 or 2010? The latter has a checkbox at the end of the data connection wizard for submit data connections - you'd like to check that if you'd want to set your SharePoint submit as the default submit data connection. You can also go to File > Submit Options (or Data > Submit Options) and set it up there, either via a single destination (email, web service, SharePoint, etc.), through rules or code.
Basically, what you'd want is to enable "Allow users to submit this form" - this will show a Submit button in both the client and browser ribbons. If you have this, then you won't need to create an extra submit button.
As for the Save vs. Submit options in the browser - saving allows users to 'temporarily' store the form in the library for further edits later before they actually submit. Saving doesn't perform actions as compared when actually submitting the form (workflows for example might not trigger when a form is only in a 'Saved' state but I'm not that certain).
Personally, I always disable the Save option in my forms simply to avoid version conflicts - for instance a user editing and submitting an old form stored in his/her local machine, submit overwrites a more recent version in the library --- you got data loss right there! You might be interested in Qdabra's DBXL Side-by-Side feature - here's an XTP with documentation for more details on that.
Hope that helps!