Here in the UK, we have one of the lowest social mobility out of all developed countries. This poses a problem whereby the rich are kept rich and the poor are kept poor, regradless of the full potential of the children and means that children do not have an equal opportunity in life. Private schools pick out the best of teachers and put them all in one pot for the rich, meaning that those you have money not only gain an unfair advantage to their poorer counterparts, they also deprive them of better teachers by milking the best ones for themselves. In turn what this means is that there is a massive gap in the quality of teaching between poor and wealthier families, with poor families often receiving significantly lower quality education and are placed at an automatic disadvantage and this is proven by the fact that privately educated children are on average much more likely to go to university and the fact that most Prime Ministers have come from the same school (eaton*ahem*eaton).
Poor social mobility means that the country is not run as well as it could be, because the people who go into higher positions are often taken from a smaller pool of people; people from the same wealthy families, breeding an elitist society in the higher positions and a larger class divide.
This larger class divide also breeds a higher crime rate. Statistics have shown that countries with smaller class divides often have lower crime rates and Australia is a prime example of this.
Banning private schools and giving everyone a more equal education will put everyone on a more even playing field. Children will no longer have their opportunities in life determined by the income of their parents, but more by their abilities and their hard work. It will shorten the gap between the rich and poor and the class divide, which in turn will reduce crime and prevent an elitist society forming at the higher positions. Of course welathier families can still give their children better education via private tutoring, but i think abolishing private tutoring is a step too far and inpractable. But i think the greatest thing of all it will do for society is given all children more equal opportunity in life, something i feel a lot of people have been in short of.