Although Administrative Closeout will occur on every project, Contract Closeout occurs only when a formal contract has been written. For instance, there may be no formal contract if you are doing the project as part of an internal system for an organization. In this case, it is unusual to have a formal contract.
However, if you are a delivering organization outside of the one where the project is being done, then it is likely that you will have a contract with specific terms and specifications. This means that you will have contract documentation that you will need to conform to in order to fulfill the obligations as written in the contract itself.
The well mentions procurement audits as a part of Contract Closeout. A procurement audit is "a structured review of the procurement process from procurement planning through contract administration." Procurement audits are prevalent in government contracting, although more and more private businesses use procurement audits on their projects. Usually, procurement audits are not controlled by the project manager but rather are conducted by a separate individual or organization.