What's more important in choosing a PhD program, advisor or institution?
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.