Independent or private schools set their own budget and do not receive any funding from the state, so they are free to employ teachers and pay them according to their own policies although most are on a par with the pay received by teachers working in the state school sector.
The main difference for a teacher looking for employment is that it is not a requirement to have the same teaching qualifications if you are thinking of working in an independent school than if you were working in a state school. This is a useful factor if you have a degree but have not undertaken any post graduate training to get qualified teacher status especially if you want to teacher older children and are therefore specialising in one particular subject. It is however common for those teaching in independent pre-preparatory and preparatory schools that cater for children from three years of age to eleven years to have a teaching qualification of some sort.
If you enjoy working in the independent school sector, then career progression is good and probably more widely available than in a state school. It is not unusual for a teacher to be taking on more responsibility after a couple of years teaching working towards the role of head of department or housemaster.