Public schools are funded by tax revenues while independent schools (also called private schools in the U.S.) are funded primarily by tuition payments, usually from the parents of the student. They also may be funded by religious or other organizations and grants.
Here in Kentucky, public schools are generally draw students from larger areas, such as an entire county, whereas independent schools are generally located in specific cities. For example, my county, one of the largest (land wise) has one public school district and two independent school districts. According to Kentucky's laws, students may attend public or independent schools irregardless of their residence in the school's district; however, independent schools do have the right to expel students who are consistent behavior problems.