Ken Starr was an INDEPENDENT Counsel. (brought in from the outside)
Mullet was a SPECIAL counsel. (he had been and was from the inside; had conflict of interests and never should have been appointed)
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The formal name for a special prosecutor, in this sense, is a "special counsel." With the title come broad investigatory powers, allowing Former Director Mueller to look into the Russia matter with limited oversight. Special counsels tend to come from within the Justice Department itself -- many are U.S. Attorneys, or assistant U.S. Attorneys, but in cases where there is deemed to be potential for a conflict of interest, someone from outside the Justice Department may be appointed.
When most people think of the position, they probably think of Ken Starr and his long investigation into President Clinton, which eventually resulted in a Congressional impeachment trial.
Here's the rub, though: Starr's investigation was conducted through the Office of Independent Counsel, which no longer exists, since the law establishing it expired in 1999.
That law, passed in 1978, was a response to Watergate and created a separate prosecutor office to avoid conflicts-of-interest questions by having an independent counsel appointed by a panel of judges from the U.S. Court of Appeals for the District of Columbia. But again, it has expired.
The new regulations under special counsels are currently appointed refer to the position as a "special counsel." When the function was created, in response to the Watergate investigation, it was known as "special prosecutor", and the individual was appointed by a judicial panel. The only person with the power to appoint an individual in these circumstances is the Attorney General, or in this case, the Deputy Attorney General, because of Jeff Sessions' recusal.
If critical evidence was found in the case but it was in, say, Russia and therefore unobtainable, "then it would be improper to seek an indictment. Critically, the entire investigation would then remain secret. It would be a violation of law for a prosecutor to make public the results of a grand jury investigation that did not result in an indictment."
So a special counsel could find incriminating information and keep it secret forever. Alternatively, a special counsel, in an effort to justify its efforts and expenditures, would have a tremendous incentive to get a prosecution even if it's unnecessary or unwarranted.
https://www.cbsnews.com/news/robert-mueller-fbi-special-counsel-what-you-need-to-know/
