PTA is a registered trademark of PTA, and you have to pay to use it. PTO is just a way of having a parents association without violating the trademark and without having to pay the annual dues.
My neighbor is a retired high school Principal and he says parent organizations can be a blessing or a curse. Boils down to whether their efforts comply with the schools needs, and sometimes, within the law.
In his case, he hoped they would support new computers in all the classrooms, and new equipment for the science lab and textbooks.
They didn't.
They put their fundraising efforts behind lights and bleachers for the football field. For decades, 3 neighboring high schools shared a stadium with lights. Never an issue, other than the PTA parents wanted their kids to have their own stadium. Well, that triggered a HUGE outcry from neighbors, a zoning battle with the county, a court battle to remove a restriction put on the school site when it was built that it never have a stadium let alone a stadium with lights. He officially was neutral, but spent the next 3 years having to go to neighborhood meetings, planning commission meetings, county board of supervisor meetings, and court hearings......to go on the record saying the school had no position, even though he personally felt there were better things to spend the money on.
3 years later, the PTA group won, the bleachers put in, the lights put up, and the kids still didn't have their computers, science equipment or textbooks.