My understanding is that dejar means "to let" and permitir means "to permit."
Either of them can carry the meaning of "to allow," which is to say that they can be used interchangeably as follows:
No dejar que los ninos juegen.
No permite que los ninos juegen.
The preposition que and the subjunctive juegen are used to say, "Don't let the children play."
Are there constructions in which dejar and permitir are used DIFFERENTLY, and cannot be interchanged?