The answer to this question is governed by 49 U.S.C. 40125 and, according to a recent FAA Legal Interpretation, is determined on a “flight by flight” basis based on a number of factors. Section 40125(c) sets the operation-specific criteria for determining public-aircraft status.
For example, if the aircraft is chartered to provide transportation or other commercial air service to the armed forces, then the Secretary of Defense must designate the operation of the aircraft as being required in the national interest in order for it to be considered a public aircraft operation. Similarly, operations under contract with the military may be considered public aircraft operations. However, according to a March, 2011 Notice of Policy Regarding Civil Aircraft Operators Providing Contract Support to Government Entities (Public Aircraft Operations), unless the FAA receives a declaration from the military or other government entity that flights under the contract are considered valid public operations and will only be operated within U.S. airspace, it will consider the operations civil operations.
Why is this distinction important? Because civil operations are governed by and must comply with the FARs, but public aircraft operations do not need to comply with the FARs. Thus, public aircraft operations are not required to comply with the operational limitations or maintenance requirements of the FARs nor are they subject to FAA enforcement for failure to comply with these FARs during public aircraft operations. However, in order to ensure that the FAA will recognize their public aircraft status, public aircraft operators will need to ensure that they comply with the applicable statutory requirements and with the FAA’s policy on public aircraft operations.