I would like to add a couple of other possibilities to supplement Chakra's list.
a) Coaching. The privates have a lot more flexibility in attracting the best coaches and keeping them. There is no way that Ezell or CPA would have won state in 2000 and 2001 without Dave Pack and Coach Smith. Both of these people can take fairly good athletes and make them great. USJ have a good program because of Marley and of course on the Public side, Satterfield made Trousdale a dynasty in the 90's. As we have seen, the money is not always the deciding factor on whether a coach stays, we know Dave Pack had a great offer from MBA and decided to stay, I'm sure Satterfiled could also find a great job if one day he decides to leave Hartsville.
At most Privates, competitive sports are strongly encouraged (By both parents and teachers) for all students even for those who have giftings more in the academic area. It would be very rare for a Private School with more than 200 High School students to have trouble in pulling together a squad in any sport.
c) Practice Time: Despite what most anti private people think, most of the kids live just a few minutes away from the school so the commitment to attend practice is a lot easier than some kids who live in a very rural environment where they may have to travel 20 or more miles to get home.
I would not put fan support as a public/private debate. I have seen great levels of support on both sides. In fact I think the Public may have a slight edge, Collinwood and Hartsville are buzzing with excitement on Friday nights where the whole town shows up.