So ive tried to find the answer myself and it seems their is some debate about it but the most popular answers i have seemed to find is that a schooling fish just likes the companionship of its own kind and the whole saftey in numbers thing so they just travel together and hang out. A shoaling fish would be a fish that schools in order to mimic a larger fish to fool predators into not eating them. they will also swim in perfect harmony basicly mimicing every movement of the fish next to them. But idk and i have been wondering this for a while now.
Thats my understanding, cories shoal, tetras school. The group that swimming in formation is a school. But school is often used loosely to refer to any group of fish, even a mixed group. Its just not the most "correct" or specific usage.
A large group of aquatic animals, especially fish, swimming together; a shoal.
definition of school from free dictionary. It uses shoal as a synonym of school. I suspect this is a case of general usage being less specific than the usage by and for scientists in a specific field.