Okay, I went to private schools K thru 12, and I taught 3 years at a public middle school. I will try to go through pros and cons of both, although my public school info only reflects what I saw and experienced at a middle school level, not elementary.
Public:
Pros: Generally larger population, therefore (in theory) more children for your child to have a chance to bond with; Exceptional Child programs (Special Education as well as "Gifted & Talented"); More teachers, so more chance to do something about a personality issue with a teacher; HOMEwork is generally discouraged (or was at the school I taught at) giving kids with other responsibilities a bit of a reprieve
Cons: Generally larger population, therefore more possibility of "middle of the road" kids (not trouble, not academically outstanding) getting lost in the fray; teachers are more likely to teach to the end of the year exam, which has a tendency to dull the passion of teaching and learning; HOMEwork is generally discouraged which leaves most practice limited to in-class (not good for classes like math, which I taught)
Private:
Pros: Generally have high levels of achievement for majority of students; smaller setting, so (in theory) more members of staff/faculty will know you and your family; almost all will have some sort of accelerated programs for advanced children; fewer changes in your kids classes from year-to-year (I graduated 8th grade with 25 of the same kids I had started kindergarten with and we were all in the same class every year) so you can get to know the kids and parents of those kids better; generally fewer major discipline issues; UNIFORMS (or at least a nice dress code; also many public schools have started making a move to uniforms); if they have teacher tenure, it is a more rigorous process than that at public schools
Cons: COST; many are parochial (associated with a particular religion, like Catholicism); there ARE kids who will be at those schools who come from money and ARE snobs (although there are just as many if not more whose parents are middle-class and make a lot of sacrifices to send their kids there); smaller populations, potentially making it harder for your child to find "kindred spirits"
Personally, I favor the private school setting. Academically, I see more opportunities at private schools than at public (obviously, there are exceptions, but as a general rule....). Socially, many will argue that a private school setting is too limited, although I don't feel like I came out of my K-12 years any less exposed to social constructs than anyone else. I DID come out with almost no experience with drugs or alcohol, but not for lack of availability. That just wasn't the group of people I hung out with. (I was a nerd, not a cool kid... :) ) Private schools ARE expensive. But if cost is an issue for you, TALK to them. See if there's any kind of payment plan arrangement that you can set up. I know a friend of mine whose mom sent all four of her kids to our gradeschool was on a payment plan with the school. Sometimes, she had to talk with them and make adjustments to the payments, but they were willing to work with her.
HTH, and best of luck with your decision!!