They're commonly brought in to investigate internal matters. Such as, financial records of a company where someone is believed to be embezzling, or a government entity official that may have engaged in corrupt practices. In this case, a special investigator may be called in to investigate allegations of impropriety on behalf of the Carrol County Police or ISP in the investigation of the Delphi Murders. They typically have unique skills or knowledge on a specific area.
There is no "special investigator" when a homicide investigation takes to long. The "special investigator" are homicide detectives, and they're usually involved from the beginning.
Believe me or don't.