Some educators use the terms interchangeably, while others insist clear distinctions exist.
Evolution of the vocabulary and current usage
Thirty years ago, people did not commonly speak of independent schools; they called them private schools. Most at the time were older, established institutions and perceived by some as elite and exclusive. (It's worth noting that until the last 1800s, most schools in Canada were private institutions; only at that time were public schools formed.) The number and types of private schools have mushroomed in the past couple of decades. As groups of schools came together to form membership associations and set standards, the term independent school came into popular use.
Governance and operations
Today, a broad range of schools forms part of the private education landscape and inquiries about governance and operations typically yield the following understanding: Schools may be owner-operated or headed by a board of directors and may be not-for-profit or for-profit. They are funded by tuition and contributions from fundraising, charitable sources and/or endowments.
For-profit and not-for-profit
Private schools are generally understood to be for-profit organizations established and controlled by one or more persons. Independent most often signifies a not-for-profit school that is accountable to a board of trustees, which operates at arm's length from the administration.
Schools referred to as independent often point to their accreditation by a peer review body. Five provinces where independent schools receive public funding require that they be recognized as accredited by a government body. But in Ontario, for example, independent schools share no across-the-board accreditation.
More information on the difference between private and independent schools