The resistance is separate from the republic.Like I said, one of the minor flaws is that they never explain why it's seemingly just the resistance fighting the First Order.
Maybe The Republic just didn't take the First Ordet setiously, the saame way the empire underestimated the rebels in the first movies.
Anyway, now that the First Order has destroyed the senate, I'm sure we'll see the Republic in subsequent films, though if I had to guess, it will be a situation where they are against the First Order but not on the side of the heroes either. Borsk Feyla situation like the EU.