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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 deje que los ninos jueguen.
No permita que los ninos jueguen.
The preposition que and the subjunctive jueguen 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?
Dejar has lots of meanings: let, abandon, quit, lend, allow, release... As you can see in the definition, the third meaning is:
tr. Consentir, permitir, no impedir.
So it means, literary "permitir", in its sense of "to not prevent" or "to allow". In this case, they're synonyms, and as such, fully interchangeable.
Obviously, with other meanings, they are not interchangeable.
All that said, I should add that, at least in Spain, "permitir" sounds more formal, and so "dejar" is more used in the everyday language.
yes, for instance, dejar de fumar. This is a totally different meaning, as you probably know already.
In this case it means "to quit", not "to allow"
First about the example:
The first sentence has an euphony problem, a more correct way to write it would be: No deje que sus niños jueguen. The second one: No permita que sus niños jueguen.
Now for the question: The verb dejar also means to let as you posted, but with the other sense of let, when you bring something and you are letting it there, which is not the case for permit, for example for the sentence "let the book in the table":
CORRECT: Deja el libro en la mesa. INCORRECT: Permita el libro en la mesa.
With that meaning you cannot interchange.
