What is the difference between polygyny permission and permission to marry from the religious affairs office?

They are two different steps. Polygyny permission is an order of the Syariah Court allowing a married man to marry again. Permission to marry is administrative approval by the Registrar of Marriage, Divorce and Reconciliation at the religious affairs office, which is needed for every marriage. For polygyny, you usually need to obtain the court order first, then take it to the religious affairs office with the marriage application form and other documents. Only then are the date of the akad and the officiant set. The sequence and form names may differ by state. See polygyny procedure and the state-by-state guide.