Speaking purely to my experience and observations of family members and close friends, I prefer teaching at a public school. Why? Because I have a union's protection, quite honestly. I grew up watching my parents teach at private schools (different kinds of private schools even), and they and their teacher friends were given so many responsibilities in addition to their paid classroom positions that they always felt like they weren't doing anything as well as they could.
This was my experience when I taught for a bit at a private Catholic prep school. The teachers were excellent, and worked so hard, but the demands outside of the classroom "burned" a few of the younger ones out especially.
And yes, the pay is an important factor. Teachers work hard. When we aren't in the classroom, we are thinking, prepping ... there is always something to do! Public schools often pay higher, and that, of course, eases other types of burdens.