I’m not sure I quite understand the question. The Ancien Regime refers generally to the monarchy of the early modern period in France before the Revolution (also sometimes referred to as the First Empire). It’s is literally the old/previous order— or what the Revolution overthrew.

There was no Ancien Regime after the Revolution. There was the Restoration Monarchy and then the July Monarchy, but the these monarchs, despite being part of the same genealogy are not monarchs of the Ancien Regime.

I think is sounds like your real question is on monarchal power? Or is it state power— as in which French state was more powerful or prominent? In terms of its potential military capabilities, or imperial expansion, or economic position?

The Ancien regime ruled as an early modern state whose networks of power were heavily dependent on patronage and privilege. Though minister like Colbert helped create some of the hallmarks of a modern bureaucracy the monarchy still essentially outsourced a number of functions (like tax collection) that modern states do not. The Restoration/July Monarchies ruled over what was functionally a modern constitutional state (though it was still very much in transition). In neither period did the monarchs rule with absolute power or authority— monarchs in both periods were constrained by different levers of power— whether the patronage networks of nobility, or elected bodies, or constitutions (written or unwritten), or the military.