In Return of the Jedi, the Rebellion defeats the Empire so the Republic is reinstated.
So, why in The Force Awakens does the First Order refer to the Resistance as “rebels”? Shouldn’t they be in power?
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As noted in that Sci-Fi exchange post, and by the Wikipedia entry for the First Order, the gist is that after the Rebellion struck a major blow to the Empire in the Battle of Endor (plus some other conflicts after that, such as the Battle of Jakku), the Empire was fractured into several splinter groups, of which the First Order is one.
Specifically, the First Order is led by former Imperial officers who adhere to the principles of the former Empire's system of government. The Resistance is a group fighting against them.
Given this, I'd say that the Resistance is a group of fighters in/from what is now a section of space left to the splinter groups of the old Empire who are fighting against the efforts of the First Order, and the Republic has been accused by the First Order of aiding and abetting the efforts of the Resistance.
Often two sides of a conflict have different ways of seeing the "truth". For instance, look at how Israel views Palestine as a part of them versus how Palestine views Palestine as independent (more or less), or how the People's Republic of China considers Taiwan one of their territories, while Taiwan considers themselves independent. I'm simplifying some complex political situations but my point is that it comes down to perspective. Even if the Republic was reinstated it is unlikely they would just instantly gain complete control over the galaxy just by destroying the second Death Star. The empire would still hold large parts of the galaxy, and it's probable that whatever the actual balance, The First Order has not accepted the rebel victory to be legitimate, and considers the Republic to be a false state in rebellion against their legitimate rulers.
According to canon Visual Dictionary by Pablo Hidalgo, the following is the geopolitical situation in TGFFA:
Resistance is nearly completely separate from the Republic. It's a private group, run separately, located separately, AND with the separate goals from the Republic, whose only connection to the Resistance is that they offer some rare grudging support.
New Republic made peace with the First Order, and is more concerned with
taxaton of trade routes than with ensuring the Bad Guys don't Strike Back - they think they won the war fully.
Resistance, on the contrary, thinks the war is not won, First Order will attack again, and are fighting against them.
Basically, Resistance, and not the New Republic, picked up the mantle - functionally and ideologically - of the Rebellion as the force opposing the Sith/Empire.