The reason why I called them the Trade Federation of Neimoidia rather than THE Trade Federation was to underline that they represent the trading interests of the Neimoidians only (we don't see any other races working for them after all).

The Republic could be like a Federal UN, a super state formed from other states which have internal constitutions, which allow for monarchs etc but where they have a shared constitution for the larger group.

Neimoidia is a member of the Republic and it's Trade Federation is doing it's own thing (like BP hiring mercenaries).