Private schools may pay their teachers at different rates. Whether they are the same as those in the public sector depends on what is being asked of you. Consider the whole package. How many hours of contact time are involved? How large or small are the classes? Is there any element of compulsory after-school work? More important, what pension arrangements are in place? Many private schools do pay into the Teachers' Pension Scheme.
It is easier to move between sectors nowadays, but schools may still prefer teachers with experience of a particular type of school. A lot depends on how well you fit the person specification. If you are switching sectors, prepare for the interview question "Why you are moving?"