The subtle distinction between a private and an independent school has to do with the legal structure of each. An independent school has a truly independent board of governors or trustees. A private school can theoretically be part of another entity such as a for profit corporation or a not for profit organization such as a church or synagogue.
The State of Utah Office of Education offers a typical definition of a private school:
"A school that is controlled by an individual or agency other than a governmental entity, which is usually supported primarily by other than public funds, and the operation of whose program rests with someone other than publicly elected or appointed officials."
McGraw-Hill's Higher Education site defines an independent school as "nonpublic school unaffiliated with any church or other agency."


