[cozzyd]

The school does matter in a lot of ways, but Stanford and Berkeley are going to be roughly equivalent in those ways.

[exmadscientist]

At least for the academic career track, the school only matters inasmuch as it connects you with your advisor. Your advisor's name, or research group's name, is what people will judge you on.

[cozzyd]

There are other factors, like the quality of your peers, facilities and colloquia, security of funding, teaching load, culture, etc.

And a high reputation school will help in grant applications and so on, where you don't get to include your advisers name.