Quite honestly, I see very little difference between the U of C and Princeton, on the whole. It depends very much on what your son wants to study. Maybe Princeton is an Ivy and the U of C isn’t, but that hasn’t stopped people applying to Stanford in droves. Until I read some of the posts here, I wasn’t even aware that the U of C was somehow thought to have less snob value.
Any school that has been a home to professors such as Milton Friedman, Cass Sunstein, Barack Obama, and many others is world-class. Perhaps I’m not as clear on the broader picture (future economics/political science major speaking), but it seems to me that the U of C should be every bit as prestigious as Princeton.
Your son should decide based on where he thinks he’ll be happier. This is a choice about where he intends to spend about 1/20th of his life, so it isn’t to be made lightly. If he likes the bright lights of the city, Chicago will probably hold greater appeal than Princeton. If it’s an idyllic campus he’s looking for, Princeton could be that place. If one school is particularly strong in his chosen fields of study, that could be a difference-maker.
Ultimately, there are 10-15 schools in the US that are truly exceptional, and the difference between those schools isn’t the quality of the education you can get, but rather the degree of self-congratulation that parents who care too much about such things will indulge in when their son/daughter is admitted. You haven’t told your son that Princeton is the only option worth considering, so you don’t appear to be one of those parents whose chief motivation in this process is the wish to say “My son got into (Ivy X).” The University of Chicago and Princeton are both among the 10-15 schools mentioned above. Almost everyone ends up loving the college they attend, and with any luck your son will be no exception. As you make this decision, know that there’s really no wrong choice.