in my part of the world

the "institutions specialized in teaching music" (meaning music schools) offer students individual working time with a teacher.
so, the basic differences (i was considering when i asked this question) would be: with your own private teacher, you negotiate everything with the teacher (naming: works, progres tempo, methods of teaching, etc.); with an institution, there is an authority higher than your piano teacher: the school, represented by a commission who can decide to kick you out if they think you don't work hard enough, or for some other objective reason - and this can be done regardless of your teacher's opinion. also, the institution provides you with theory/harmony/forms/etc. teachers, and with an environment in which everybody plays an instrument. also, a major difference is that with a private teacher, you can get away with lesser achievements, but an institution will not allow it. (unless you have money or connections

)