Original post by SarcAndSparkBy teaching in a private school, you're perpetuating a system of inequality, so there's that.
Other than that, private schools can appoint whoever they like- you don't need QTS. This is the case for some state schools too, but most prefer teachers with QTS. Some private schools like post-grad qualifications but this isn't universal. If you can offer an extra curricular (e.g. sports coaching) this is desirable to many private schools even if it's nothing to do with your subject.
Most private schools stick to the teacher pay scale used in state schools, although they don't have to. In many private schools, however, this is pro-rata'd to the shorter term times- so your take home pay will be lower BUT you will have longer holidays. Equally, if you have to do Saturday school, it may be Pro-rata'd to take this into account. Some private schools have different rates of pay- usually this is made clear in the job advert, but if you are unsure do ask at interview. Some private schools are struggling financially at the moment, so may try to get away with low rates of pay. Some will pay you more if they want to attract you to their school.
Working hours and workload are broadly similar but with a different emphasis. At a private school, you'll likely have smaller class sizes, so maybe less marking, but you'll usually be expected to do certain extra-curricular activities e.g. run a club, attend various parents events. A lot of private schools, especially at secondary level, have school or school events on Saturday mornings, so you may not get a full weekend. Workload depends a lot on the school and there's probably more differences between individual private schools than there is between state and private.
One positive, is that you usually don't get asked to teach outside your specialism in private schools, but this is becoming increasingly common in state schools. Even at upper primary/prep level, you're likely to have some specialist teachers, which is nice if you don't enjoy art/sport/music/science etc.
One big negative is that private schools are far more likely to close than state schools - especially at the moment. If a school stops being financially viable to run, then that's it, and you could lose your employment quite suddenly. In state schools, if you're on a permanent contract, it is really unusual to be made redundant (although it can happen). Obviously this doesn't apply to all private schools- most of the big names are very financially secure, but if you're taking a job at a school you've never heard of before (especially a tiny prep) it's worth looking into their finances a bit.