The Allied Worlds is like a combination of NATO and the European Union. The individual planets within the alliance have their own militaries and legal systems, but the central government on Yih can and does oblige members to contribute to mutual defense, as with the Peacekeepers. Economically the alliance is heavily integrated, with a single currency and free trade agreements between members.
The basic divide in Allied Worlds politics concerns how much power the central government on Yih should be given. Centralists favor more concentrated authority while autonomists favor a looser alliance of more independent states. This divide trickles down to smaller divisions of power as well, between planetary governments and the administrative regions that are the remnants of past sovereign nations.
The pull for autonomy is strongest on Yih and Sweetwater. In Yih's case the provinces that were once independent countries dating back to before the space age agitate for independence. Sweetwater is better thought of as two states, with the wealthy benthic cities being the real beneficiaries of the alliance. Most of the surface, except some of the larger fixed islands, is a de facto no man's land of isolated atavist tribes and roving seafarers both piratical and peaceful. You can divide the surface dwellers into those that willfully ignore the AW government and those that don't even know what the AW is in the first place.
Besides the debate over centralized authority, the big issue dividing AW citizens is the Partisans. The AW's original raison d'etre was to protect the member planets in the event the Partisans tried taking over the system. One side favors deeper economic ties with the Partisans, while another wants to treat them as a threat. Both sides acknowledge that the Partisans are problematic, but differ on whether deepening economic ties would make them less authoritarian or just feed the status quo. There are also a pawful of tankies who insist that Partisan Territory is a paradise and Firefly did nothing wrong.