They both check you don’t have a criminal record and/or aren’t prohibited from purchasing guns. They both allow you to buy a firearm once they have cleared. You have to pay either way. Purchasing permits you get from the police, background checks are done by licensed dealers but the authorisation comes from executive power as well.
What’s the difference?
EDIT: My good faith is still on less shaky ground than the UN saying guns cause violence but having armed guards whenever and wherever they like.