Outside of the physics community, there may indeed be some differences in reputation, though I'd expect them to be small, since anybody hiring physics PhD's will have a rough knowledge of Berkeley's excellent reputation in physics. Sure, in other fields (or for undergrad), the reputations may not be equivalent. NB: I'm Stanford undergrad, MIT physics PhD, so I have no personal reason to hype up Berkeley (the opposite bias, in fact...).
Some data: Last I knew, Times Higher Education surveyed published tenured professors worldwide about the best schools in their field. I think it's probably the best and mabye only good data.
Every year, six schools have been head and shoulders above the rest overall (in random order): Harvard, MIT, Oxford, Cambridge, Stanford, and Berkeley.
I think OP should go with the better PI, however suggestion Stanford and Berkeley have the same level of prestige seems wrong to me