Both great schools. As others have mentioned, your PI is going to matter a ton, there are really bad PI's who will deliberately keep you a few extra years, and there are good ones who will let help you and progress your career. It's sort of a gamble as to which is which unless you know them personally.
One other consideration though is that Berkeley is the safety school for people who don't get into Stanford. If all else is similar, I would go to Stanford, as it will provide a lot more opportunities down the line than Berkeley.
> Berkeley is the safety school for people who don't get into Stanford.
This is true for some undergrads, although there were plenty of people who got into both and chose Berkeley.
But it's definitely not true at the Graduate level. It just depends on the program. Some programs at Cal are better than at Stanford, and vice versa. But in make cases the two schools collaborate quite a bit. It's not unheard of for someone at Stanford to have a Cal professor as an advisor for example, or the other way around.
I think as you move further from the specific work your lab did, school brand name matters more and more. Berkeley is a public school, and it’s going to be more closely associated with other top public schools like UW-Madison, while Stanford will be associated more closely with private schools like MIT.
I think people in the Bay Area forget this, but as you move further from the bay Berkeley is regarded more and more like another public school. I’m not saying it’s fair, but it is just another public school, where Stanford is a globally recognized school.
> I think people in the Bay Area forget this, but as you move further from the bay Berkeley is regarded more and more like another public school.
As someone who lives far from the Bay Area: This is ridiculously wrong - especially when applied to PhD.
It all depends on the discipline you're in. And when it comes to engineering, most of these top public schools outrank most of the Ivy's. Almost no one cares if you got your EE PhD from Princeton or Yale. They definitely will care if you got it from Berkeley. Or Ann Arbor. Or Illinois, And so on.
Reminds me of a coworker who went to Princeton for his Mechanical Engineering PhD who deeply regretted it. He was coming from abroad, and naturally assumed an Ivy would be better. He really wishes he had gone to one of the Midwest public schools instead. He hadn't heard of any of them before coming to the US (and clearly hadn't bothered doing his research before applying).
Multiple brand surveys have shown that Berkeley is one of the best known brands outside the USA after Harvard, right alongside Stanford. Especially in Asia.
I assure you Berkeley is also globally recognized (disclaimer: no links to either school).
EDIT: I'd also personally would want to stay as far away as possible from people who would see a PhD from Berkeley as a lesser one (based on general "prestige").
One other consideration though is that Berkeley is the safety school for people who don't get into Stanford. If all else is similar, I would go to Stanford, as it will provide a lot more opportunities down the line than Berkeley.