Hoo boy, I created a long answer... sorry you missed it. ;) I guess I can't do that. And now it's gone.
I homeschooled my children until my oldest was in fifth grade. He's about to graduate high school. He was a much more confident child back then. I found all my children were more self-confident before public school. They have all asked me to return to home-schooling. Mostly because they don't like dealing with bullying and idiots in school.
Homeschooling focuses more on academics, tailoring more to the specific educational needs of the child. It is more likely that a child's weaknesses will be determined and strengthened while homeschooling than in public school. Public schools might be aware of academic strengths and weaknesses through testing, but they lack time, staff, and financial resources to focus on individual needs.
It is possible to involve homeschooled children in group instruction, activities, and sports. I had my children in a state-funded charter school that required 18 hours of teacher time. I hired teachers who divided their time into group classes or who taught individually. I was able to hire teachers who taught academics and piano. My children were in karate, tumbling, and swim lessons. All of my academic supplies and curriculum were paid for. (It was great!)
That brings me to the downside of homeschooling. It can be expensive. I moved away from that school system and no longer had that advantage. I open-enrolled part-time in the local school system for a year using the public school system curriculum, but it did not have the same advantages of teacher support. In fact, I was required to hire a teacher to oversee my student progress for the year and I could not afford this, which is why I ended up putting my children into public school the following year.
The other con, for me, was managing children of varying ages and varying topics. It can be difficult, especially when I had a toddler and a baby demanding my attention (and housework piling up.)
Homeschooling requires a lot of support. I was basically trying to do it all myself with five children in the end. I was disappointed to have to give it up, but I will always support the option. In my opinion, homeschooling provides a great foundation for children.