my take is:
private: smaller enviroment, better education, more attention and support to the student, more work, more independant kids, upper class childhood, higher expecations, you can get expelled and suspended for stupid stuff, you have a bigger reputation
public: bigger social life, parties, diversity, chillness factor, school spirit, easier, less attention, more independance
boarding: closer friendships, mini-college, harder work, solid reputation and legacy, dorm setting, roommates, school spirit