How so? It looks like the same Starfleet to me. It would certainly be odd for the producers to intend the prequel series to feature some other group called Starfleet Command.
^Functionally speaking, Earth Starfleet became UFP Starfleet. Although one ended and the other began, it may have been nothing more than a name change in the organizations rapid evolution.
Well, for one thing, Earth Starfleet isn't military, but the Federation Starfleet is. That alone should tell you something.
I agree that when the Federation was formed, its Starfleet must have taken cues from the Earth version as to how to organize things. But logically speaking, they can't actually be the same organization. Totally different areas of jurisdiction and responsibility, let alone mission profiles.