Hi. I did the move five years ago. Behaviour is less of an issue. Parents, however, are more of an issue. Because they are paying they are more demanding and frankly can be rather ridiculous at times. You spend a lot of time carefully handling parents. There is less acceptance that their child is in the wrong and the school (particularly in one where numbers are an issue) will bend over backwards to appease them. Do keep an eye on numbers. Many privste schools are struggling and are not awash with money - this then increases the appeasement level. The anount of work then that staff do, in essentially giving x child one to one tuition in lunch hours really ramps up.
There is a lot more homework given and this has to be turned round really quickly.
There will be evenings, weekends to work for open days, founders day, etc, etc and that is unpaid. Always a three line whip to attend. Be very wary of boarding environments - they never tell you how many extra hours they want you to do at interview.
Don't think there won't be safeguarding issues, etc, there are and sometimes more difficult to deal with. Likewise SEN issues can be difficult because it is up to the parents to accept there is a difficulty and pay for an assessment and on occasions they just don't want to know.
Generally, though, I prefer it. Not having the level of behaviour significantly reduces paperwork, however, that time is taken up with help with students - which is a good thing. The students are generally lovely, vthere's a free lunch - not to be underestimated because it saves loads of money; you get more time to actually teach. Things like cover are generally more and forget ideas about not doing displays - all of that stuff has to happen as often there are fewer TA and no cover supervisors.