Verified answer

They should have the same powers, but civilian law enforcement would have broader jurisdiction. Military police would have jurisdiction over military personnel, where the civilians would have jurisdiction over civilians and military.

I believe Civilian Law enforcement has more power under the legal system, but have no example to give. I would also be interested to hear from MP's about dealing with crimes committed on base? I know that at Fort Hood it was civilian law enforcement that finally ended the shooting, and they respond to active shooters on the bases in their jurisdiction. I also know that if a member of the military is arrested they can face discipline through the military as well as the civilian courts without violating double jeopardy laws.

Military stays within it's own turf but civilian's turf can be policed by military in emergencies, it's rare since the USA is not policed by military officers as normal thing. Could happen if we got invaded.

We could say military has more power if the two went head to head, since the military is for war and combat on the large scale with more powerful weapons. The civilian federal government might throw tear gas at them in response.

Neither is 'more powerful' than the other. basically, they have much the same authority, albeit in different areas. Each has jurisdiction in certain areas but not in others.

They all have the same "power" but within their respective jurisdictions.