I think it depends what you mean by brightness:
Light intensity at the 'focus' which determines the 'throw' (distance from flashlight at which light intensity in the centre of the beam is at a specific level).
Or total output (in all directions) which maynot be useful if you want 'spot/distance' use.
On top of that in different usage different lights can appear 'brighter' especially if they have different tints.
IMO though my Raw NS Cree (on freshly charged battery) is 'brighter' on high than the P1D-CE (on primary - I have no rechargeable CR123A, and dont know if an RCR would be brighter - I know there are regulation 'issues' with RCR) on high, but not by much, and the tints are different (my Raw is blueish, and the PID-CE is more yellow/softer). Don't have a Raw Cree, so can't compare.
I am sure you will get some better answers from other (more knowledgeable) users, and maybe even some 'numbers' to compare.
Rob