Q: "let's" is a short form of "let us", but does it have the same meaning? When is it better to say "let us" and not "let's"?
A1: They carry the same general meaning. Let us, though, is more emphatic; used for emphasis, and/or used formally.
A2: "Let's" is commonly used in modern English when you say:
Let's do something...
eg. Let's...play football, go to the cinema, have a pint down the pub, etc.
No one uses the original "Let us" form in this context anymore unless they are aristocracy!
eg. "I say old chap - let us go boating upon the river this afternoon what what!"
"Let us" is still used but in a different context and to mean "allow/permit us".
eg. allow us...to do something
...Let us in (to the house), Let us help you, etc.
"let" we all know how to use
"Let" me borrow a pen.
"Lets" is to show action referring only to one thing.
My brother "lets" me hold his car.
"Let's" is a contraction word meaning let us.
"Let's" go to the mall. "Let us" go to the mall