The problem is that I can give equally strong reasons for the public side. We really do have a great public school program. Here's what I see as the other side.
1) We aren't Catholic.
2) JROTC -- I know he'd love it, I worry about how much encouragement there is for them to choose a military career. The military is 100% out for my kid, because he's got a medical condition they reject for.
3) Arts -- My son has been involved in chorus and drama at his school and loves both. Our public has lots of different choruses, and a theater arts "academy" where they can take up to 2 classes a year for 4 years, with lots and lots of choices like IB Film, and theater production, and of course acting and television productions classes, plus lots of productions every year. Private has good instrumental music and good visual arts, but only 2 chorus options, one audition based ensemble, and no drama classes. They have 2 stage productions a year, at least one of which conflicts with the sport he wants to play.
4) Languages -- private offers Spanish and French. Public offers many more including Arabic. My son has said for several years that he'd like to study Arabic in high school, and study abroad in that part of the world in college.
5) Social Sciences -- this is my son's strongest subject, and there are many more high level classes at public, more AP's etc . . .
6) Sports -- program at private is definitely stronger, but that also means that a lot of kids go there specifically for sports. I worry about playing time for my kid who is talented but average height and plays a sport/position where size counts.
7) Money -- this should be first. I think the tuition workable for us, but we'd have to cut a lot. If he went public my son could do some summer programs on college campuses plus sports camps, drama camps etc . . . He could continue to be in the ski club (it's middle school/high school mixed). We could take trips as a family. If he chooses private all those things would need to be cut.
Aha..the plot thickens..you do bring up some good points. What does your son say about these counterpoints? Has he thought about this side of the coin? Maybe he has and those issues are not as important to him as the ones in your previous post. Hard to say.
In regards to the arts issue...from what I understand kids need to take basically 4 yrs of each of math, english, science, soc studies, and foreign language for college entrance, plus a couple years of phy ed and maybe computers, health, public speaking, etc. required by the high school. I don't think there is that much time for too many electives anyway. On the con side though, 4 years of religion is probably required at the private school, which means one less opportunity for other electives. But again, on the plus side, religious ed is very much like social studies in high school, which he would enjoy.
In regards to arabic...what if he tries it for a year and doesn't even like it? Hard to choose a school based on one class offering. There is definite value in that class if he intends to work in the middle east eventually. Otherwise spanish is the most practical choice for work in the US.
Truly, there are rebuttals for every issue, and then those can be refuted too. You will both need to decide which set of reasonings matter more.
Being a bright kid, she got with the program eventually, but it seemed strange to see someone so gifted who could actually believe she didn't need to associate with people who weren't "up to her level". When her brother reached HS age, his dad moved him to the public school 
