Actually, no! These terms are often used interchangeably, but they all mean different things.
Nonprofit means the entity, usually a corporation, is organized for a nonprofit purpose.
501(c)(3) means a nonprofit organization that has been recognized by the IRS as being tax-exempt by virtue of its charitable programs.
Tax-exemption is the result of a nonprofit organization being recognized by the IRS as being organized for any purpose allowable under 501(c)(3) - 501(c)(27).