I de inglés

	1	2	6	7	24	29
cases	-O	+O	+ING	+OI	+ONC	+RI
TO ALLOW		✔️	✔️	✔️	✔️
TO LET				✔️		✔️
TO PERMIT	✔️	✔️	✔️	✔️	✔️

2 We don't allow that sort of behaviour here. 
6 We don't allow smoking in the theatre.
7 I'm sorry but we can't allow you to take that book out of the library.
24 They don't allow dogs in the hotel.

In all these examples, permit could be used instead of allow, but would always be more formal. In formal contexts, the passive construction is common (see permit).
Allow with the gerund (case 6) refers to general prohibition; where there is a personal object the infinitive construction is used (case 7).
🌱 Noun: allowance, money you assign someone regularly.

7 She didn't let the children go out alone.
29 Let yourself go!

Note the form of the infinitive after let (without ‘to’).
Let has no passive form, so case 7 in the passive would become «The children
were not allowed (permitted) to go out alone».
Let us (Let’s) is a more positive form of suggestion than ‘Shall we?’. ‘Let’s have a party’ encourages other people, where ‘Shall we have a party?’ is asking for advice or support. Note that the tag question in this case would be: ‘Let’s have a party, shall we?’

☣️ Problem: Let me come in! (NOT leave)

🎵 But note: Leave me alone or Let me alone (Don’t bother me!).
Here both verbs are acceptable.

1 Play will begin at 2 o'clock weather permitting.
2 Ball games are not permitted in the park. 

See allow, where permit could be substituted in formal contexts in cases 6, 7 and 24.
🌱Nouns: permit, official document, and permission, allowing you to do something (uncountable).

Permit (noun) is pronounced with stress on the first syllable /ˈpɜː^rmɪt/
The verb, permit /pə^rˈmɪt/, and the noun, permission /pə^rˈmɪʃ^ən/, are pronounced with the stress on the second syllable.