A wise person once said, "Never let school get in the way of your education."  I was home schooled up until I started college, and now I've almost finished my Bachelor's degree. The contrast between the two experiences has been enlightening. Although I tend to be more favorable towards home schooling, both environments have their advantages.
Freedom. My dad introduced me to programming when I was 10 or 11, and in 9th and 10th grade my learning accelerated a lot in that area. It was easy for me to spend lots of time on computers and programming because we chose the curriculum. Anything that was educational could be a core subject. I got to pour lots of time into learning about things I loved. I started to care a lot about my education, and I became ambitious.
Flexibility. This goes hand-in-hand with freedom. I learned about computers in spurts. When I thought of an idea for a program to write or some other project to do, I would work on it many hours each day for two or three weeks. During those times, I stopped working on other subjects. After I completed the project, I then would catch up in math, science, English, etc. My schedule adapted to fit my needs instead of the other way around. 
There was also flexibility on a larger scale. I took geometry in 9th grade. It came easily to me, so I finished it quickly and did algebra 2 in the same year. The next year I was able to go right into pre-calculus. This was completely natural because I didn't have to be in sync with a class; I could go at my own speed.
Efficiency. I learned programming from the internet (mainly through Google). I learned math from watching short DVD lectures and then working through problems on my own (eventually I started reading the teacher's manual instead of watching the lectures because it was faster). I watched video lectures from a man who loved history, and he made it a great experience. There are many ways to disseminate information, and home schooling made it easier to do it in the most efficient way for each subject. In addition, learning alone is often easier for me. I could stop and think about things that don't make sense or speed through the parts I already got.
This experience was largely undiluted by homework assignments or exams which often consist of busy work and aren't beneficial to learning. Formal education tries to teach you stuff, but it also tries to verify that you've learned that stuff. Or at the very least, it tries to make sure you've expended a certain amount of time and effort in activities that are related to the subject material. The effort to provide verification often hurts the teaching process, but I was mostly free from that in home school. I had to take a standardized test from California every year, but it was easy.
Social interactions. I never spent a ton of time interacting with peers until I started my mission. I developed a lot of close friendships, many of which are still active since many from my mission came to BYU. I also had tons of fun. I never had that kind of social experience while I was home schooled. Being at a university with lots of friends has been great too.
To be fair, my lack of a social life before my mission was probably due mostly to me being heavily introverted and living in a rural area. But being part of a formal education system does give you more opportunities to be with friends.
Actually, I think meaningful social interactions are the main inherent advantage of a formal education system. Books and the internet just can't replicate conversations with other people.
Exposure. A structured system with required courses makes sure you get a broad education instead of becoming too specialized. I learned a lot of good things from my university courses that I probably wouldn't have chosen to learn about otherwise.
Accountability. As much as I hate homework and exams for reasons stated above, they can be helpful to make sure you actually do stuff. There was accountability while I was home schooled; it was just less structured. I liked the self-motivated, independent nature of home schooling, but I think it's also valuable to be able to function in a system with harder constraints. Deadlines can certainly be helpful when you're lacking motivation.
Here is my grand vision. It's still pretty general, so you'll have to mentally fill in the blanks on your own for now. I've purposefully left some details not figured out yet because I'm going for the big picture.
Imagine a high school or university where only half of your time is spent in classes. You would be expected to spend the rest of the time on educational projects that were largely up to your own devising. These projects could be all sorts of things, and they could be large (like learning to make a website from scratch) or small (like reading a book and writing an essay about it). They would allow you to branch out and explore topics that might interest you. They would also let you dig deeper into topics that you already like.
At a high level, the idea would be to take the euphoric home school experience I've described and simply encapsulate it within a formal education system. The school could somehow provide opportunities for students to connect with other students that are interested in similar topics. Teachers would act more as mentors and less as lecturers. They would also provide accountability because the individual projects would need to be approved and given a final grade by a faculty mentor. 
This connecting with peers and mentors is worth emphasizing. I'm not talking about simply doing group projects where you divide up the task evenly and get it done as quickly as possible. I'm thinking of a less formal type of interaction that stimulates and motivates/inspires those involved. If you've read the 7 Habits of Highly Effective People, the thing I'm describing is called synergy. It's hard to define (it's been a while since I've read that book...), but I think most people have experienced it at one point or another. It's the feeling you might have when staying up late, talking to a room mate about an exciting idea you just had.
This is the big thing home school lacks. University could also do a lot better at it—being buried in course work all the time isn't very conducive to those kinds of experiences.
This hybrid system would have the advantages of both formal education and home schooling in addition to extremely meaningful social interactions that are currently lacking in both systems. It would give students a more diverse experience: they would learn to function with hard constraints, but they would also learn how to excel when it's all up to you. They'd have flexibility because course work would take up only part of their time instead of all of it.
But most importantly, this system would help students gain a deep appreciation for learning. That would give them the drive to accomplish great things no matter what system they find themselves in down the road.
 This is the format I always use when quoting myself.
 In fact, I'm currently putting off school assignments so I can finish this essay. This is only possible because I'm at the end of the semester and I've already completed all my major projects. I usually don't have this liberty.
 There is precedent for these kinds of projects, at least in university. I've been able to get six credits for working on a research project of my own devising, and it's mentored by one of the computer science faculty. But that's only 5% of all the classes I have to take. The new system would make these experiences a major part of the core curriculum.