I have worked in various schools- both independent and state over 20 years. I would research the school thoroughly as the grass isn't always greener.
In my experience, some private schools can act like they own you- they can chew you up and spit you out. I have seen this happen many times. Yes you will get much smaller classes, with a TA and specialist teachers which is great for PPA time. However, don't underestimate how pushy some of the parents can be. They are paying and expect you to deliver- be prepared for reams of demanding e-mails! SMT tend to side with the parents as they don't want to lose their custom/reputation. They are first and foremost a business.
If you teach in the Prep School you may have to be teaching/running after school activities until 7pm. At my last school, you had to be ready to receive children in your classroom at 8am. SEN provision/support can be lacking too. You might need to teach on Saturdays as well. Much longer holidays are a bonus, but after long jam packed days you need them.
Look very closely at the management of a private school and who owns it. If the classes are very small, then you may be out of a job very quickly. One school I taught in literally closed down over night- all the staff lost their jobs and children had to find schools very quickly.
Not all private schools offer staff discounts and pay can be worse.
I am not anti Independent schools by any means- my children attend 2 brilliant private schools and are very happy. There are some fantastic schools out there. However I much prefer teaching in the state sector. Class sizes are much larger and holidays shorter, but I feel more supported and am paid much better! Resources for my subject are also far superior to the private school I have just left.
It really depends on the individual school- try and talk to someone who works there. If they have recently advertised numerous jobs- avoid like the plague!