What is the difference between/Cual es la diferencia entre "Allow / Permit / Let"?

Question

Como saber cuando debo de poner Allow/ Permit / Let

Answer 1

These can all have the same meaning, although they have differing meanings as well.

Allow me to help you. = Permit me to help you. = Let me help you. = Déjame ayudarte.

To allow and to permit, generally, are synonyms.

To leave in this case means the same as dejar, but in can also mean (in a different context) the same thing as salir.

Answer 2

Como ya dijo Jeremias, muchas veces las palabras «allow» y «permit» se usan como sinónimos. Es posible, pero es menos frecuente, que la palabra «let» se use precisamente de la misma manera. Por ejemplo, fíjase en las diferencias entre las siguientes frases:

Let me in:

?Let me in
?Allow/permit me to come in.

Let me do something

?Let me see it.
?Allow/permit me to see it.

También, hay muchos casos en los que solo se usa la palabra «let»:

Open the door so an animal can go outside

?Let the dog out. (Open the door so the dog can go out)
?Allow/permit the dog outside.

to release something

?Let go of my hand (Release my hand)
?Allow/permit go of my hand

Math

?Let A = B
?Allow/permit A = B

To fire someone (i.e. give someone the sack/dismiss from employment)

?They let me go. (I was fired)
?They Allowed/permitted me go.