A. I have thought about this a lot. And, yes, a lot has been written on it. I will try to be succinct. I am relying mainly on my own knowledge and experience, Webster's 1828 and 1913 dictionaries, and current mainstream dictionaries.
 
As with so many words, there is considerable overlap between school and education (both the noun and verb forms). Without including all the caveats and conditions, here I go, and maybe this mini-thesis will be of use.
 
Over time, it appears that schooling and educating have increasingly merged in their meaning. On the one hand, meaning the same thing does not help us in knowing how to use them. On the other hand, the merger of meanings substantiates what I will say after I give some possible definitions.
 
School: A place or organization outside the home where teachers instruct, teach, or drill students (i.e., children and youth) in specific knowledge or skills such as reading, language, mathematics, and arts and, allegedly, only secondarily in manners, philosophy, and morals.
 
Education: The bringing up and instruction of children and youth to enlighten their understanding, instill their philosophy, develop their morals, form their manners, correct their tempers, give them knowledge and train their skills such as in reading, language, mathematics, and arts, and fit them for usefulness in their families, associations, and communities. Education comprehends all that series of instruction and discipline which is intended to accomplish the aforementioned.
 
People tend to think that state/public schools only school children, but they also educate them, always and at all times. That is, philosophers of education recognize that all schooling is education. All schooling is the teaching, training, and indoctrination - to imbue with particular opinions, points of view, or principles - of children and youth. The impression that many parents and the public have and that many public-school teachers and advocates present is that school is mainly or only about instructing students in knowledge and skills and that it does not teach them in ways of sectarian values and beliefs, worldview, and how to judge or evaluate things according to philosophical presuppositions.