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I am not very sure about the difference between these two words.

I looked them up in Oxford and Longman Dic, they also said that let's is the simplified form of let us, but they neither told me the usage of let us.

I think:

Let us go to the restaurant=Please allow us to go to the restaurant.

Let's go to the restaurant=Shall we go to the restaurant.

Is that right?

Thx for ur help.
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Comments  (Page 2) 

I understand it the same way as you do "Let us" is "Permit us"/ "Allow us" as in "Please let us go to Andy's slumber party."

"Let's" is a proposal or a suggestion as in "Let's go to the beach this evening".