I see a definite difference with Jack, who's in school not by his own choice. I homeschooled him one year and it just didn't work out at all. He's an extrovert and I'm an introvert, and we came up with a schedule that wore me out but left him understimulated. Plus, I really couldn't afford to keep paying for all the music classes, archery classes, etc. In public school, music lessons are free. Then there's the issue of my health- I seemed to neglect it when I had him home; something that didn't happen when I had Hannah home with me.
Yes, keeping a very extraverted teen stimulated is quite an undertaking. I'm glad you found a solution that helps you stay healthy!
Most of the families we know who are homeschooling teenagers have them enrolled in lots of different lessons, and are on the go practically every day of the week. In our case, even when we find free or low-cost things to do, we can't afford to keep enough gas in the car to be driving somewhere every day. So we are homeschoolers who actually spend most of our time at home and in the neighborhood, which seemed to work fine for dd1 when she was younger, and still seems to be a good fit for dd2 now -- but dd1 started expressing a need for a broader canvass, and eventually really spelled it out to us that the only way she could get the kind of experience she wanted was by going to school.
Overall, I'm really glad she's had this experience of getting to the root of her own discontent, articulating her needs to us, and setting some major changes in motion. She went from saying that "nothing in my life ever changes" to seeing how much power she really had to affect her own destiny, and that has to be a good thing.