I'm one of those teachers who a long time ago did have a studio on my own and worked at a music school. I've also had parents tell me that they would rather just take from a private teacher and not be at a music school. I've got feelings on both! However, before I continue, I do need to mention that all music schools are not created equal! And, all private teachers are not created equal. This is more of a generalization of both.
I will say that it is hard finding teachers who want to teach at a music school simply because they don't make as much money. However, on the plus side, they don't have to deal with the business side of things. They simply show up and teach! This can be amazing for teachers who simply love to teach and don't want to do billing, advertising, and organization. More than likely, teachers who work at a music school really LOVE to teach and are there because of that. Teachers also gain a sense of community within the school and have comradery with the other teachers that they would not have solo teaching at home. This is perfect when they need to have their student perform for another teacher! All they have to do is knock and ask! It is obviously wonderful for the students as well, because it's built in performance practice, and another teacher's ideas that they get to hear!
For parents, taking lessons at a music school can almost be "one stop shopping". If there is a teacher that is not a great fit for the student, makinging a shift to a different teacher is sometimes possible. Often at music schools, there is the possibility of multi-instrument instruction, so you don't have to go out and hunt for a different teacher. The other plus side is that if you trust the music school and its mission, you are comfortable with their teachers and know that they are all "safe" individuals.
Most individual teachers don't have a commercial space. Going to a "business" that is primarily for music lessons definitely has a different feel than going to your teacher's house. Now, I went to my teachers house for years and years, and a lot of my students have come to my house for lessons, and there is absolutely nothing wrong with it! However, when I show up to teach at our studio, I'm focused and ready to teach. It's true that a lot of my students felt that my house was more of an "intimate" space- but it was also a little more noisy (my kids or husband or dogs). There seems to be a little more structure in my lessons at the studio vs. my house. It feels like a better learning environment- AND there's not a pantry full of food to raid or laundry you can voluntarily fold (LOL).
The best thing about a music school? It's the community- if it's the right community for you and your family. Hopefully, it's nice being able to keep the financial part separate- knowing there's a person to talk to about that and that oversees the instruction of the teachers (my students are not allowed to comment!) There are different teacher personalities and all kinds or opportunities to take part in. There's variation in classes and performances/recitals. There's a higher standard.
That in a nutshell is the difference! I love sharing this unique perspective and hope it was helpful. If you have thoughts, comments, or questions, don't hesitate to reach out!