Dr. Courtney said:
I respectfully disagree. There are not many opportunities to impress folks outside of one's home institution, and the REU adviser's letter of recommendation can boost grad school applications. While it is hard to compare institutions with this question, there is a fairly straightforward way to compare how well the actual research mentors/advisers are known in the field.
Look up the two candidate advisers on Google Scholar. See which one has more citations. If one has 10k citations and one has under 1k citations, one is definitely better known in the field than the other.
Of course, this assumes the student can do a good enough job in a short REU to merit an excellent recommendation from each adviser. Now, I don't steer students too much based on this consideration. Picking an adviser and a project that one is excited about is more important. But I am very honest with my own students. If they have a chance to work with someone much better known in a field than I am (say, 10 times the citations I have), I encourage them to jump at the opportunity.
Not that citations are the best way to discern how respected different scientist are in a field, but it is probably the easiest to apply and describe simply in this kind of discussion.