These words have similar meanings. Permit is more formal than allow.
- Smoking is not allowed / permitted.
Both allow and permit can be followed by object + infinitive.
- We do not allow / permit our kids to stay up late.
- We do not allow / permit people to smoke in the house.
- Please allow / permit me to go.
An -ing form is used after allow / permit if there is no personal object.
- We do not allow / permit smoking in the house.
Permit can be used in the passive form with it. Passive forms of allow cannot be used with it.
- It is not permitted to smoke in the house. (NOT It is not allowed to smoke in the house.)
With adverb particles only allow is used.
- She wouldn't allow me in. (NOT She wouldn't permit me in.)
Let is less formal than allow and permit. It is followed by an object + infinitive without to.
- Please allow me to get you a drink. (Quite formal)
- Please let me get you a drink. (Rather informal)
Passive structures with let are not usually possible.
- I wasn't allowed to go. (NOT I wasn't let go.)
Let can be used with adverb particles like in and out.
- She wouldn't let me in.