I'm sorry for the formatting. I don't know why it became one big line.

They are mostly equivalent. These mean the same thing: I let him go. I allowed him to go. Notice that the verb after "let" doesn't have "to" while the verb after "allow" does. In more advanced terms, there are some differences. "Allow" can be used in the passive voice, for example, while "let" cannot. I am allowed to go. [No equivalent with "let." "I am let go" is meaningless.] "Allow" can also stand on its own, while "let" cannot. We allow pets. [Correct, meaning "pets are permitted."] [No equivalent with "let." "We let pets" is incomplete. It would need to be something like "We let pets stay here," or "We let pets come inside."] You could express the same meaning, however, as: We allow pets to stay here. We let pets stay here. There are also some idiomatic uses where they are not equivalent. I will let you know. [Meaning, "I will give you an answer soon."] I will allow you to know [Meaning, "I will permit you to know.] Dave