Let me start with the obvious. Home schooling is an objectively deficient form of education. It inhibits the development of life skills, such as for negotiating social institutions and employability. It undermines political community, such as by preventing children from learning society's common sense and dividing them from citizens of different homes. It provides a lower general quality of education since its 'teachers' know nothing more of what or how to teach than their textbooks tell them.
It is not surprising that home-schooling is especially attractive to those adults, notably religious ones, for whom these characteristics are actually desirable. These parents do not want their children to develop into free adults who can decide for themselves what to make of their life. Instead, they want to cultivate an asociality so that their children will relate only to members of their own community; an inability for critical thought so they cannot question the dogmas they are fed; and helplessness to function in the mainstream economy so that they cannot even dream of escape.
This use of home schooling is morally odious. Indeed it constitutes a kind of child abuse. Parental rights include not only control rights over how their children should be raised, but also responsibilities for using that control to the benefit of the child and its development and flourishing.
Consider even the seemingly most successful case of home-schooling, John Stuart Mill, always cited in these discussions despite occurring before public schooling was a real alternative. On the one hand Mill junior could read Plato in Greek before he was 10 and he went on to become England's premier public intellectual. On the other hand the character of that education was a model of what happens when parental rights are understood as a license rather than as a trusteeship. Mill senior considered his child a suitable object for scientific experimentation, conveniently powerless to resist having ideas from Bentham's utilitarian philosophy tried out on him to see if they would work. That education scarred Mill for life and caused him a nervous breakdown at the age of 20, which he only recovered from by reaching beyond his father's curriculum to poetry and music.
However this is not the end of the story. There are many people who are attracted to home-schooling and related alternatives (private schools that select on the basis of parental religion) not because of their failings but in spite of them. Many public schools, especially high-schools, are large, soulless, officious, heavily guarded institutions that treat students as problems to be contained until they are old enough to be released rather than respecting them as equal human persons in their care. The education they offer is a cruel joke, more like training in the 19th century idea of work as pointless, alienating, compulsory drudgery than the upliftment of the soul and the opening of opportunities.
Public education is a central responsibility for any society. When it fails, it drives morally responsible parents to an objectively deficient form of education, undermining the future of a society and its citizens and providing cover for those morally odious parents who actively desire that. On the other hand, the flourishing of public education removes the justification for home-schooling, and makes its central odiousness clear.
One can thus compare the two systems in terms of their ideals. A well-functioning, properly motivated and resourced public education system is superior to home-schooling in all but the most exceptional cases. When public schools work well teachers are committed subject-specific experts trained in best practises for reaching different kinds of learners; students become familiar with the fact of pluralism (cf Rawls) and learn skills and ambitions that expand the kinds of lives they have the real freedom to achieve; and so on. These are functions that home-schooling cannot reach even at its best.
Parents have a duty to help their children develop into autonomous individuals who can go out into the world and decide what to make of themselves. What we call education is central to that. If public schools are failing and parents believe they can do better, or better for their children's special needs, then home-schooling is justified as a superior means to the same end. But parents don't have a moral right to home-school their children based on their own specific value scheme, for example to try to turn their children into little acolytes of their religion. That goes against their children's right to a life of their own. Home schooling is an inherently deficient form of education that is morally permissible only as long as public schooling is even worse.