No unions or bureaucracy
Public school teachers are unionized as a rule. This has its good points and its bad points depending on who you ask. Private school teachers are not unionized. As a result, you will need to be flexible and adaptable. Basically your job is whatever the school determines it to be depending on the exigencies of the moment and the day. One important consideration, however: if you are accustomed to telling an administrator "That's not my job." then perhaps you will need to look elsewhere for a teaching job. In a private school flexibility is critical. Everybody pitches in to get whatever needs to be done accomplished.
Another difference is that you won't have to deal with layers of bureaucracy to get simple things done. Private schools are generally small, close-knit communities where everybody knows everybody. Getting something fixed or some supplies replensihed is not ordinarily a big deal. Floating a new idea and bringing that idea to fruition is generally easier in a smaller school setting. Not so many people to convince.
Students who want to be there
Students attend private school because their parents want them to get a first rate education. The admissions process can be quite rigorous involving as it does testing and interviews to determine if the student is a good fit for the school and vice-versa. As a result, you won't be dealing with children who have issues with being in school or learning for that matter.
The other reality is that classes are generally small. 12-15 students is typical. Because the students have been carefully screened and selected, they are usually quite capable of doing serious amounts of academic work. Furthermore, because the school hired you with a degree or two in your chosen field or subject, you can run with your exploration of the subject matter. Naturally you will do it within the parameters of the school's curriculum. But be assured that your students will accomplish in a year what many students in other academic settings would take two or even three years to accomplish.
Private schools consider the educational process a partnership of three entities: school, student and parent. Since the parents have made a conscious decision to send their children to private school and are spending significant amounts of money to do this, they expect to be involved.
The typical day school will have parents helping with a host of tasks and events which will make your work a lot easier and more effective. Parents will also expect to meet with you regularly to discuss their children's academic progress. Expect a lot of interaction with parents in a private school.
The school and the parents will expect great things from you. If you don't deliver, your contract will not be renewed. And there will be no recourse to a union when that happens. Simply put: you chose teaching because you love your subject and you enjoy teaching it to others. Keep the fires of passion for your subject well-fanned and make sure all concerned are turned on by how you teach and what you teach.
Teaching in a private school offers you an opportunity which could well turn out to be your life's work. That is so unusual in this age of constant job changing. Find a school which appreciates you. Then do what you love doing: teaching young people.