I have significant experience of teaching in the private sector (as well as in maintained schools) in London and elsewhere and would underscore some of what has been suggested. You will probably be expected to take a wider role in an independent school, with other minor duties not often undertaken in the state sector.
Private schools vary in quality, though not as much as the variance between state schools. Teaching in a poor private school will have its challenges, sometimes quite surprising ones. Pay at less good schools will be similar to maintained pay scales, possibly less but that is rare.
The best private schools can be rewarding places to work but they will expect their 'pound of flesh' from you. You will be better paid (significantly more) but competition for jobs at these places will be fierce from high calibre candidates with Oxbridge/Russell Group university backgrounds.
If you are looking at teaching in the private sector best to start applying for non-dept head roles. You need to be realistic about your own experience/and or qualifications.
Teachers I worked with who made the transition to the private sector sometimes struggled to adapt and left shortly thereafter. Others flourished right from the off and went on to great things. A lot depends on you. Teaching in the private sector is no sinecure: you'll be expected to be good at what you do and produce results. Pupils' parents will be supportive but only if you're doing the best for their offspring. Parent/teacher evenings at the better private schools will be a revelation to you.
I have a freind who left our school for the private sector and she absolutely loved it. She was a classroom ICT teacher at our school and went to a private school as Head of ICT. She was sacked as soon as she had sorted out their schemes of work, as she had had an 'unsatisfactory probation period'. The amount of work she put in to organising that department to be sacked after 6 months was appalling.
So you do have to remember that there are different terms of employment in private schools than there are in state schools, you don't necessarily get the same pay and you can be sacked without anywhere near as much paperwork.
I have been teaching 8 years (secondary English) and an second in dept at a large challenging London school. I have taught in similar schools throughout my career. Now keen to become head of English but wasn't successful in interviews last year in London state schools (I had three). Feeling increasingly confused about what I want and tempted by a move across into private sector, under the (maybe misguided) impression that it will be easier, both to get and to do the job. Please, lovely mumsnetters, tell me- am I deluded? Would you advise this, or not?