I've had a brief experience with private schools in Palo Alto at Elementary school level. It was great. After touring various options (Super academic, relatively laid back, montessori, catholic etc) - I decided on a hybrid model between Montessori and structured academic. My experience was very positive, most importantly my child loved it and we got treated like royalty by the school administration (I suppose everyone does).
We decided to move my child to public because we wanted to give it a shot. I must say that the private curriculum is ~ 1 year ahead of public. This is uniform across most private academic schools, and this is their chief value proposition - that they keep your child on an advanced track. However, unless you take solace in the fact that your 1st grader can do 2nd grade math or higher, IMO in big picture it does not matter.
In public schools, the teachers are very supportive to the student's individuality and ability. Public schools are well funded. Also, the parental participation in public schools is phenomenal.
So barring the ego massage that private schools will offer, I see little difference between private and public. Private academic schools are decidedly more aggressive, and for some, it can be attractive. However, for me personally, there are no regrets to moving to public!