"Let somebody do something" and "allow somebody to do something" convey the same basic meaning, but they have slightly different connotations and are used in different contexts.

"Let somebody do something" is more informal and can imply a sense of permission or allowance. For example:

I let my younger sister borrow my clothes" or "He let me use his car for the weekend.

"Allow somebody to do something" is more formal and can imply a sense of granting permission or giving someone the opportunity to do something. For example:

The teacher allowed her students to use their phones during class.

The law allows citizens to vote at age 18.

Overall, "let" is more informal and is used in casual conversation, while "allow" is more formal and is often used in legal or official contexts.