Laz,
A very significant part of that situation is that in the private school the parents are motivated to see that their kids actually do something. Since it costs the parents something of actual value (money) they want to get the full value of it. They are also interested in their kids performance as that helps make the parents feel their money is well spent. Having parental interest in the kids education tells the kid that this is of importance to the both of them and they should not just blow it off.
It was said before that people value that which has actually cost them something to achieve. That which is merely given freely is valued for the effort it took to get it, nothing.
Having said that, while I value the concept that EVERY child in the US (I am talking citizens and LEGAL immigrants) deserves to be educated, I think providing the first 12 years for free has failed as a concept of education.
You can put a kid in a school but if the parents are not interested and the kid does not want to learn, they will not be educated.