The simple answer is “worldview.” Of course there s a lot more to it than this but this is the foundation. A group of people come together whom all have similar goals and similar belief system. This group usually has a strong foundation in the Word of God, the Bible. Their principles, people, purposes, processes, and products are all based out of the Bible or founded Biblically or function to be Biblical. Of course a lot of Christian schools are just schools with Christians in them because they error in one of many possible ways.
One of the biggest problems we see is that the kids are there for the wrong reasons. The parents, kids, teachers, and admins are NOT all on the same page. For example, the parents want their kids in a Christina school so that they are protected and not in the “evil” secular world. In this instance the kids, most of them, are only at this school because the parents give them no other choice. This kind over sheltering is actually bad for Christians and Christian schools. There is no unity. The teachers, admins, and some of the students who truly want to express their Christian faith in real life have great dreams and plans to “be the difference” and are founded Biblically. They want to grow their own faiths and challenge other students. They want to reach out to their lost friends and restore their community. But not everyone is on board. Sorry that was a rabbit trail of my own observation. I’ve spent time talking about the mistakes certain Christian schools have made, but let me say that when Christian education is done the right way, it is a beautiful thing.
Christian education comes from a Biblical worldview, so some of the obvious difference from public education is the beliefs in major social issues and scientific issues. The Bible is a book that actually can be used and believed in at a Christian school so evolution, abortion, gay marriage, and other sins (that the public school would never dare talk bad about) are brought up in the Christian school.
One of the better features of a Christian school compared to a public school is the fact that school is not limited to institutional learning of the basic classes and skills (art, gym, science, math, reading, writing, etc) but in the Christian school the students are taught about personal growth in their lives relating to their spiritual faith as well as more personalized focused on personal growth in other areas too (like physical shape, mental health, etc). In public school this care is skipped over, for the most part…too many kids, not enough time. The “personal” health of students in public school is assumed to be healthy.