Politically I think each country would run itself much as before, surely?
The central government would always want to exert some control to establish uniform standards and generally work its will in policy matters.
And regardless, you'd be paying taxes to a planetary government that you had no say in.
The United Federation of Planets extends the reach of government power, and reduces the ability of actual people to have a say, even further. It's a dream or a nightmare, depending on how much you trust distant elites to decide everything for you (what wars to fight, where to send foreign aid...).