At our school, there are internships and clinics. The major difference is that in an internship, you are not "practicing" law.
The internship involves mostly observation and research. Depending on the internship, you could have client contact, but would not represent a client. In a judicial internship, you might do research and give a summary to the judge, or perhaps write some draft orders. Some judges let you sit on the bench while they are hearing cases.
In a clinic, you are provisionally certified by the Supreme Court and actually practice law and represent clients. Thus, if you do the prosecution clinic, you actually prosecute criminals (under the supervision of a prosecutor).
Our internships and clinics are primarily with businesses and agencies in the surrounding area. I have not yet done either, but have been selected for an internship next semester and am looking forward to it. Both programs have been highly recommended as a way to get "real" experience and make connections that could lead to future employment opportunities (or at least references).