Administrative closure and contractual closure are both different steps in the project closure phase. Administrative closure usually happens before contractual closure.
Administrative closure consists of releasing the team members (who may be allocated to another project), evaluating employee performance, and patenting (when patents are needed).
Contract closure consists of acquiring full acceptance from the client on the final product. Also, in this step, the buyer (the client) has to pay all the outstanding payments to the seller (the company who made the product).