It really depend on what model the world government is. Is it a gathering of sovenrign nation-state equals, or are todays nation-states merely impotent client states?
I admit I favor the idea that the United Earth is more a council of sovereigns than a top down central authority. Reasons would be it's sound (to me) to be a more interesting arrangement, and yes more complicated, which would make for involved stories.
UE could help in the areas of international projects, disputes, resources. Also would co-ordinate inter-planetary and inter-stellar activity, arrange colonies (although nation-states wouldn't be restricted to only going through the UE). It's through the UE that the nations of Earth decide upon and send their spokesman to the Federation council.
i don't think the nation-states would "be under" the UE. None of what we saw in Enterprise requires that the nation-states of Earth to have relinquished their sovereignty, only that "something" called the United Earth be in existence.
If one of the duties delegated to the UE is to interact with off-world civilizations, then what we saw makes sense.