Allow is more formal than let, but allow and permit are used with a preposition (Allow me TO help you with that) and let is used without a preposition (Let me help you with that). Also, let cannot be used in passive voice (You are allowed to smoke in here; You are let is incorrect)

In everyday speaking they are formal and informal and most often mean the same. However Allow refers to allowing letting "someone do something" you give consent or permission "a permit" Let is not restricted to A person or someones animals or vehicle. It also means to Turn a "blind eye" Or "to not prevent" I am quiet happy to allow you to park your car on my drive. but I won't let you smoke in my house" allow and let can be interchanged here. "I will not let the dirty washing pile up" or is possible to say "allow" However I think that many of the examples you get in some dictionaries the distinction is obscure and let and allow can be 99.99% interchanged.For permission. It is only the other uses of let seen here where there is a difference between the two.. https://en.oxforddictionaries.com/definition/let "allowed to" "Let + ( ) " " Let me/it/him/her/them/those/these/our/your +( ---- )" "allow me/him/her/them +(--)" So the only real difference is the construction of the sentence for permit/permission. When using Allow or Let. And let also has other meanings apart from permit/permission. You do not have to have a preposition when using allow. Except when stating it in imperative form. "you are not allowed TO smoke" "I won't allow that" "It is not allowed" "who allowed you in here' and many more valid examples are possible without using "allowed to" Same with let "I won't let that happen" "who let you in here" "it is not let" -> = it is not rented another meaning of let

muchisimas gracias =)!!!!!! let's practice if you want to...good day !

You can use both in formal contexts and informal.