They were both handled according to the published laws and regulations in place at the time.
Kenneth Starr was appointed under the independent counsel statute -- which meant he reported to Congress, not the U.S. Justice Department.
The independent counsel statute -- which was always of dubious constitutional legitimacy anyway -- was allowed to expire in 1999. Some cynics say the whole purpose of the Starr investigation and the impeachment of Bill Clinton was to give Congress the political leverage they needed to do this.
Robert Mueller was appointed under internal DOJ rules. He operated as a DOJ official, and his reporting responsibilities lay entirely within the DOJ.
Details. Details. They don’t need no stinking details.
Details for anyone who wants them:
After the expiration of the Ethics in Government Act in 1999, the Office of Independent Counsel was replaced with the office of Special Counsel, defined by regulation 28 CFR 600, which in turn is based on Congressional statute 28 USC 510.