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.

I personally think that your rapport with your supervisor is of primary importance, and I personally didn't care to choose my PhD by university when I got the opportunity. (Assuming that both your options are offering what you want from a PhD in terms of subject and skill sets.)

As a final year student in biomedical sciences, what I have mostly found is the following:

Money

Your supervisor should be able to fund you, and your experiments fully. You should not come to a situation where you are choosing the second best option (especially in biology) because of the lack of funds. This will ruin the quality of your thesis.

Go-to guy

You should have a supervisor or a post doc in the lab who will be able to help you when you initially start out to answer your stupid questions. There will be many, and you'll need to find some one in the lab who is friendly enough, and patient enough to answer them. You'll know them when you see them. And of course, the other members of the lab do make a difference. See what they are like.

Higher up or lower down?

You should need to ask yourself, how high up the totem pole you want your supervisor. Remember, the higher up they are (especially professors) the lesser you are likely to meet them on a regular basis, and most probably have lost touch with bench experience. On the other hand, profs have better connections and their recommendation carry weight.

The argument for university preference goes like this: If your prof doesn't have his own lab, and relies on community equipment, then you must go to a university which has the money to spend. Otherwise the choice of university is mostly trivial.