You can't really say if one is better than the other. A lot of it has to do with the quality of public and private schools in your area, your family, your child, etc. There are pros and cons to both, you have to weigh out what is best for your family.
Our area has a lot of good private schools and our public school system is one of the top ranked. We went with public because we know our daughter is getting a thorough education. Her Sr. High school which is only 11-12 grades is rated in the top 1% of high schools in the nation. You can't say that for all districts.
If we were not in the Plano ISD, we would either send our daughter to private school or move to Highland Park in Dallas because Highland Park ISD is excellent. One of the primary reasons we chose Plano to live was the school system.
Secondly, another reason we chose to go and stay private is that we wanted our daughter to have a more diversified and balanced education. Private schools can be very good but it can also lead to some sheltering of the students from the "real world". I mean this as students leave the protected environment of private school to enter college, they will meet other students that are very different and they will have to learn how to get along, network, etc with ALL types of people.
I suggest you visit the schools you are considering WHILE SCHOOL IS IN SESSION and get a feel for how they are run. Also, look into the scores of students on State testing. What percentage of students go to college from the high school level.
Whatever you choose, get involved, volunteer and support your school.
That's just a couple of thought from my perspective. Good luck.