Generally speaking, the two phrases have the same meaning. However, technically speaking, prescribed fire is the application of fire in a burn unit to wildland fuels under specified environmental conditions to accomplish specific land management goals and objectives. Whereas, controlled burn is a more generic term with different meanings to different people.
The term controlled burn predates the term prescribed fire. As the science, application, and profession of burning developed and expanded, the term prescribed fire was introduced to better reflect the process involved in planning and conducting a burn.