In a public school everyone is taught the same thing at the same time from the same book(s) in the same way.
In real life, people learn in different ways at different paces and they have different interests. In a homeschool, each child learns at his own pace in his own way and, largely, he learns that which is of interest to him. If junior likes spiders, we go to the library and get every spider book. We're doing reading, biology, ecology, geography just from the topic of spiders. Wow. How does the public school cover all these topics, eh? In a boring way, I'm sure.
In a public school, girls and boys of the same age are put in the same classes.
In real life girls mature faster than boys, so the public school is mixing the mature with the iofftopicture and trying to teach them all the same material from the same books in the same way for the same amount of time. Hello! Can we say "I'm an idiot for designing this public school system!"
After a public school education, students don't remember very much because they weren't interested in most of what was taught (in part because it wasn't taught the way they learn). We can only hope that they won't need that information later.
After a homeschool education, students remember far more, because the lessons were customized both to interest and to the way the student learns.
In a public school, you can't raise your hand and get special help just anytime you need it. In a homeschool, you can, because you literally have a private tutor. Maybe this is why historically the upper class in Europe and America hired private tutors. There's a thought.
The purpose of a public school is to teach people to stop thinking for themselves, and to start obeying the master class. Observe: the questions are given to you, and the "right" answer is in the back of the book. You don't have to think original thoughts on your own (not even original questions! they're given to you!). You just have to mimic that "they" want to hear. The final product is a generation of programmed droids accustomed from about age five to obeying the master class.
Homeschooling is the opposite. Homeschooling is original science. It's what a baby does: See ball. Touch ball. Oh, ball moved! Chase ball. Touch ball again. Oh, ball moved! Repeat. Conclude: touch ball and it moves. Next, pick up ball. Smell it. Taste it. Good? Bad? Try again. Maybe repeat a third time. Conclude good/bad. Move on to the next object in view and repeat. Etc.
That's original thinking. It's good. Every single five-year-old entering kindergarden does this sort of original thinking because it's natural, but after five years of public schooling, can you find more than three students per class who still do it? I don't think so. By age 10 or so, public school students are largely mimicing the questions and answers that are in their books. They're not doing much original thinking any more.
Homeschoolers take classes at the YMCA. They do Scouts. They're in martial arts schools. They play in soccer leagues. They do all the extracurricular activities that any other child does. Consequently, they know all about talking with children their age.
Additionally, they know all about talking with people not their age, because they do not live in an artificial environment. Public school students live in an artificial environment. A public school classroom is just about the only place on God's green earth where you'll find 30 children all within 10 months age of each other. Everywhere else in real life ages are mixed from birth to age 106. Think about it: the mall, the movie theater, the library, the grocery store, the bank, the park, the office where dad works, etc. Everywhere else in real life ages are mixed, including the dojo and the YMCA sports leagues. Thus, homeschooling is actually better for socialization.
Homeschoolers score higher than their public counterparts on every standardized test given in America. Hmm. Which educational system is better?