What's more important in choosing a PhD program, advisor or institution? In terms of judging a doctoral program--and possibly also relevant for postdoc-ships(?)--which is the more important consideration, who your advisor is or the institution at which you study?

Obviously both are important, but when it comes down to it, which should weigh more heavily?

For example:

Professor Alpha is a premiere name in the field, but he's set up at Southern State University, an otherwise middling school. Perhaps he's there because he's a big fish in a small pond and gets carte blanche and limitless resources to do what he wants. His graduate students are thus treated similarly.

On the other hand, Professor Beta is a new, unproven professor at Top University, a world-renowned, leading institution. He personally may not have the resources and freedom as Professor Alpha, but he's at an institution whose name carries weight. Therefore the school certainly has the capacity for the same, if not more, resources, but access for Professor Beta's grad students will be more limited as they must be shared with other professors' groups.

Assuming all else is equally appealing (funding, location, research topics, etc), which offer do you choose, or which program would you hold in higher regard?

(The situation doesn't have to be so stark as like Harvard vs. Middle-Of-Nowhere University, but it makes the point clearer.)
Obviously one needs a competent advisor with whom one is compatible. But assuming that both professors qualify, I think what matters most is the quality of the students who will be your peers. You need to surround yourself with students who, from day 1, expect nothing less of themselves than to produce novel scientific research of the highest caliber, present it at top meetings, publish it in top journals, and forth. Ultimately you will learn more from your peers than from your advisor. A sufficiently talented and ambitious cohort will hold the bar high for you and push you to excel whereas a sufficiently talentless and unambitious cohort will help you make excuses for your own failures to reach your potential. 

In my experience, top schools with top graduate programs have the sorts of students you want to surround yourself with. Second tier regional programs may, but I have yet to see it.