What Does The Law Say About Special Counsels?

It's pretty short and easy to read.  There are ten sections, not more than four paragraphs each.  You can read it yourself - courtesy of Cornell University Law School.

28 CFR Part 600 - GENERAL POWERS OF SPECIAL COUNSEL

§ 600.1 Grounds for appointing a Special Counsel.The Attorney General, or in cases in which the Attorney General is recused, the Acting Attorney General, will appoint a Special Counsel when he or she determines that criminal investigation of a person or matter is warranted and -
(a) That investigation or prosecution of that person or matter by a United States Attorney's Office or litigating Division of the Department of Justice would present a conflict of interest for the Department or other extraordinary circumstances; and
(b) That under the circumstances, it would be in the public interest to appoint an outside Special Counsel to assume responsibility for the matter.

§ 600.2 Alternatives available to the Attorney General.When matters are brought to the attention of the Attorney General that might warrant consideration of appointment of a Special Counsel, the Attorney General may:
(a) Appoint a Special Counsel;
(b) Direct that an initial investigation, consisting of such factual inquiry or legal research as the Attorney General deems appropriate, be conducted in order to better inform the decision; or
(c) Conclude that under the circumstances of the matter, the public interest would not be served by removing the investigation from the normal processes of the Department, and that the appropriate component of the Department should handle the matter. If the Attorney General reaches this conclusion, he or she may direct that appropriate steps be taken to mitigate any conflicts of interest, such as recusal of particular officials. 

§ 600.3 Qualifications of the Special Counsel.(a) An individual named as Special Counsel shall be a lawyer with a reputation for integrity and impartial decisionmaking, and with appropriate experience to ensure both that the investigation will be conducted ably, expeditiously and thoroughly, and that investigative and prosecutorial decisions will be supported by an informed understanding of the criminal law and Department of Justice policies. The Special Counsel shall be selected from outside the United States Government. Special Counsels shall agree that their responsibilities as Special Counsel shall take first precedence in their professional lives, and that it may be necessary to devote their full time to the investigation, depending on its complexity and the stage of the investigation.
(b) The Attorney General shall consult with the Assistant Attorney General for Administration to ensure an appropriate method of appointment, and to ensure that a Special Counsel undergoes an appropriate background investigation and a detailed review of ethics and conflicts of interest issues. A Special Counsel shall be appointed as a “confidential employee” as defined in 5 U.S.C. 7511(b)(2)(C). 

§ 600.4 Jurisdiction.(a)Original jurisdiction. The jurisdiction of a Special Counsel shall be established by the Attorney General. The Special Counsel will be provided with a specific factual statement of the matter to be investigated. The jurisdiction of a Special Counsel shall also include the authority to investigate and prosecute federal crimes committed in the course of, and with intent to interfere with, the Special Counsel's investigation, such as perjury, obstruction of justice, destruction of evidence, and intimidation of witnesses; and to conduct appeals arising out of the matter being investigated and/or prosecuted.
(b)Additional jurisdiction. If in the course of his or her investigation the Special Counsel concludes that additional jurisdiction beyond that specified in his or her original jurisdiction is necessary in order to fully investigate and resolve the matters assigned, or to investigate new matters that come to light in the course of his or her investigation, he or she shall consult with the Attorney General, who will determine whether to include the additional matters within the Special Counsel's jurisdiction or assign them elsewhere.
(c)Civil and administrative jurisdiction. If in the course of his or her investigation the Special Counsel determines that administrative remedies, civil sanctions or other governmental action outside the criminal justice system might be appropriate, he or she shall consult with the Attorney General with respect to the appropriate component to take any necessary action. A Special Counsel shall not have civil or administrative authority unless specifically granted such jurisdiction by the Attorney General. 

§ 600.5 Staff.A Special Counsel may request the assignment of appropriate Department employees to assist the Special Counsel. The Department shall gather and provide the Special Counsel with the names and resumes of appropriate personnel available for detail. The Special Counsel may also request the detail of specific employees, and the office for which the designated employee works shall make reasonable efforts to accommodate the request. The Special Counsel shall assign the duties and supervise the work of such employees while they are assigned to the Special Counsel. If necessary, the Special Counsel may request that additional personnel be hired or assigned from outside the Department. All personnel in the Department shall cooperate to the fullest extent possible with the Special Counsel. 

§ 600.6 Powers and authority.Subject to the limitations in the following paragraphs, the Special Counsel shall exercise, within the scope of his or her jurisdiction, the full power and independent authority to exercise all investigative and prosecutorial functions of any United States Attorney. Except as provided in this part, the Special Counsel shall determine whether and to what extent to inform or consult with the Attorney General or others within the Department about the conduct of his or her duties and responsibilities. 

§ 600.7 Conduct and accountability.(a) A Special Counsel shall comply with the rules, regulations, procedures, practices and policies of the Department of Justice. He or she shall consult with appropriate offices within the Department for guidance with respect to established practices, policies and procedures of the Department, including ethics and security regulations and procedures. Should the Special Counsel conclude that the extraordinary circumstances of any particular decision would render compliance with required review and approval procedures by the designated Departmental component inappropriate, he or she may consult directly with the Attorney General.
(b) The Special Counsel shall not be subject to the day-to-day supervision of any official of the Department. However, the Attorney General may request that the Special Counsel provide an explanation for any investigative or prosecutorial step, and may after review conclude that the action is so inappropriate or unwarranted under established Departmental practices that it should not be pursued. In conducting that review, the Attorney General will give great weight to the views of the Special Counsel. If the Attorney General concludes that a proposed action by a Special Counsel should not be pursued, the Attorney General shall notify Congress as specified in § 600.9(a)(3).
(c) The Special Counsel and staff shall be subject to disciplinary action for misconduct and breach of ethical duties under the same standards and to the same extent as are other employees of the Department of Justice. Inquiries into such matters shall be handled through the appropriate office of the Department upon the approval of the Attorney General.
(d) The Special Counsel may be disciplined or removed from office only by the personal action of the Attorney General. The Attorney General may remove a Special Counsel for misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies. The Attorney General shall inform the Special Counsel in writing of the specific reason for his or her removal. 

§ 600.8 Notification and reports by the Special Counsel.(a)Budget.

(1) A Special Counsel shall be provided all appropriate resources by the Department of Justice. Within the first 60 days of his or her appointment, the Special Counsel shall develop a proposed budget for the current fiscal year with the assistance of the Justice Management Division for the Attorney General's review and approval. Based on the proposal, the Attorney General shall establish a budget for the operations of the Special Counsel. The budget shall include a request for assignment of personnel, with a description of the qualifications needed.

(2) Thereafter, 90 days before the beginning of each fiscal year, the Special Counsel shall report to the Attorney General the status of the investigation, and provide a budget request for the following year. The Attorney General shall determine whether the investigation should continue and, if so, establish the budget for the next year.

(b)Notification of significant events. The Special Counsel shall notify the Attorney General of events in the course of his or her investigation in conformity with the Departmental guidelines with respect to Urgent Reports.
(c)Closing documentation. At the conclusion of the Special Counsel's work, he or she shall provide the Attorney General with a confidential report explaining the prosecution or declination decisions reached by the Special Counsel.

§ 600.9 Notification and reports by the Attorney General.
(a) The Attorney General will notify the Chairman and Ranking Minority Member of the Judiciary Committees of each House of Congress, with an explanation for each action -

(1) Upon appointing a Special Counsel;

(2) Upon removing any Special Counsel; and

(3) Upon conclusion of the Special Counsels investigation, including, to the extent consistent with applicable law, a description and explanation of instances (if any) in which the Attorney General concluded that a proposed action by a Special Counsel was so inappropriate or unwarranted under established Departmental practices that it should not be pursued.

(b) The notification requirement in paragraph (a)(1) of this section may be tolled by the Attorney General upon a finding that legitimate investigative or privacy concerns require confidentiality. At such time as confidentiality is no longer needed, the notification will be provided.
(c) The Attorney General may determine that public release of these reports would be in the public interest, to the extent that release would comply with applicable legal restrictions. All other releases of information by any Department of Justice employee, including the Special Counsel and staff, concerning matters handled by Special Counsels shall be governed by the generally applicable Departmental guidelines concerning public comment with respect to any criminal investigation, and relevant law.

§ 600.10 No creation of rights.
The regulations in this part are not intended to, do not, and may not be relied upon to create any rights, substantive or procedural, enforceable at law or equity, by any person or entity, in any matter, civil, criminal, or administrative.

From what I can tell, the terms special prosecutor and special counsels seem to be used interchangeably.