It's very different. My DC were all at state primary school until year 3, then another year while we were in Canada. They also went to prep school, then another independent school, then independent secondaries. I taught in a variety of state secondaries for 4 years.
The behaviour issues tend to be different, the expectations somewhat different and behaviour management can be wildly different from school to school.
What I remark mostly is that the way of speaking to the pupils is very different between state and private. In state there is a lot more telling on the whole and a lot less communication and negotiation. The pupils have an expectation of the teacher being more in charge and a disciplinarian and less a lecturer, so perversely some tend to kick more against heavy-handed discipline. Unless pretty much every teacher in a school is using the same techniques, pupils will not look kindly on a teacher they see as "weak", ie one who treats them more as equals.
On the whole my experience of discipline in state schools is that it's a lot stricter and more disciplinarian but also perversely there are more behaviour problems. Many pupils rightly have problems with being spoken down to and told to do things in a commanding tone, but in the right context with a teacher they like will be as compliant and attentive as any private school pupil and willing to do what the teacher directs. The important thing is that that feel respected as a person, not looked down or viewed as an inconvenience.
The advantage with most private schools is that the teachers are much less harried and therefore more able to deal with pupils individually, have the full support of their SMT regarding any behaviour problems, mostly the support of the parents (although not always) and the pupils are more likely to comply by instructions- arguably because they are not being spoken to like cattle. State school teachers are very overworked on the whole and develop a carapace that does not permit individualism.
When I hear what my daughter's teachers have to put up with in her classes of 15, I feel for them. I know that in state school most of the girls would not have the relationship they have with their teachers. They are lucky that in permitting what would be seen as a lot of cheekiness in a state school, they are getting the best out of each other and really managing to pass on their knowledge.
Teenagers are teenagers anywhere- there's not really much difference between state and private school ones (apart from the better resources in most private schools). The difference imo is in the handling of them.