In the US, a separate public transit system specifically for schoolchildren exists, usually because there is no effective public transit system for everybody. A child cannot catch a city bus from their home to their school when no city bus ever comes within 3 miles of their home.
Europeans often don't understand, because American urban planning makes it seem like one would have to be a madman in order to do things that way. It's not madness. That's just the way the economic incentives aligned in the past, and no one has cared to re-break the bones in order to set them properly, so everyone is crippled from birth by the history of the infrastructure.
Some school systems in the US are subject to judicial desegregation orders, so that the racially-divided neighborhoods do not result in segregated schools. These may require transit routes that are not necessarily coincident with the daily work commute. Some systems also have magnet or specialty schools. Some have even greater ability to choose (or win via lottery) a school other than the nearest to one's home. As a result, the commute for one child could be longer than that for their parents, and using the regular public transit could result in hours on trains and buses. In such cases, a school bus would essentially be an express, with no stops between school and the residential neighborhood.
And, of course, once you run one school bus, that means you now have a school bus infrastructure, which makes it easier to run more buses.
The public transit system is usually designed around moving people between workplaces, mainly in the city core, and homes, which are more evenly distributed. Schools are also more evenly distributed, as villages, towns, smaller cities, and unincorporated townships may have had one or more of their own before being annexed into the metropolitan city. As a result, the nearest transit station to both home and school may be the same station. Or they could be on lines that run nearly parallel before meeting in the downtown hub.
One possible solution to this would be to build a large, unified school near the city's central transit hub. And that would likely be selected by an algorithm as most efficient. But politically, people prefer their schools to be near their homes. And separated by grade levels. With athletic fields and playgrounds. And lots of separate and distinct administrative regions. Again, the technically sensible solution is untenable, because people just want what they like (and what makes them money).
Someone above said ~25$/student/day: sure that's higher than the average commuter fare right now. Couldn't a unified system serve everyone better?
And encourage students to spend at least a bit of the day not locked into an institution designed around them.
Also in the US once you are outside a medium to large city public transportation may not be available at all. Or is limited to systems which run only during commuting hours and are designed to transport people from a few centralized locations into the closest city.
I'm pretty sure that when i was at school in the UK, there were specific school bus routes. They weren't operated by the schools or the local education authority, so i assume they were special services run by the local bus operators, using stock that would be used on other routes at other times.
I believe each route served multiple schools - kids from my school would share the bus with kids from other schools. Since we were the grammar school, full of nerds in garish purple blazers, and they were, well, not, this was sometimes a stressful experience.
It may be relevant that this was in Colchester, a town with numerous schools, some quite close to each other, whose catchment areas take in lots of small villages, where there isn't necessarily a good normal bus service.
Mind you, i got the (ordinary!) train, so i'm not sure of the detail.
I could alternatively have taken a (public transport) bus, but the fare would have come out of my pocket (i.e. out of the small monthly allowance I was given to spend on whatever I chose).
It's a system with alot of history, impact and strong feelings on both the schools and the city itself.
Children who live too far from the school to walk catch a bus, an ordinary scheduled bus on a standard route, or the train# or the metro. They get special bus passes (some get them free, I think most now).
What I'm saying I think is why on earth is the Boston school system in the public transport business to start with?
(#) Doing your homework on the train on the way to school is one ritual I was amused recently to see is still going strong. I did this and my dad did this (pre WW2).