Firstly, most people forming a private foundation will prefer to form a 501(c)(3) public charity mainly because the benefits are greater. Specifically, the main difference is a person donating money to a 501(c)(3) public charity can donate up to 50% of their adjusted gross income. With a private foundation, a person donating money can only do so up to 30% of their adjusted gross income.
The dividing line between being deemed a 501(c)(3) public charity or a private foundation is based upon the sources of support of the foundation. In other words, if the foundation’s sources of support are from a relatively few number of people then the sources of support will cause the IRS to deem the foundation a private foundation. If the sources of support are large enough, the IRS will deem the foundation to be a 501(c)(3).