The Rebellion was just a rebellion, not even remotely affiliated with any government, which was clearly established. In TFA, they mention that there is a relationship between the Republic and the Resistance, but don’t establish what that relationship is.
As far as I can tell, the Republic and the First Order are two separate states which co-exist uneasily in a sort of Galactic Cold War (at least up until General Hux presumably starts a war with a Pearl Harbor-style surprise attack). The Resistance is the Republic’s covert ops arm inside the First Order’s territory. Kind of like the French Maquis during WWII.