Allow can be followed by an object + an infinitive, or an adverbial particle. It cannot be used as a passive structure. eg. She allows him to watch the television every evening. He allows his daughter out until 11pm at the weekend.
Permit can be followed by an object and an infinitive but not by an adverb particle. It can be used in a passive structure. It is slightly more formal than “allow”. Eg: They are not permitted to enter the building after dark. Football is not permitted on the grass.