"Let us" can be used to add gravitas: "Let us not forget the widows of our fallen heroes" has a less casual tone than "Let's not forget the widows....." Let's play football, let's have a drink, let's watch a film....are all casual. "Let's" is spoken, "let us" is written.

To aupport his understanding, a friend send me this text: "A commonly used English contraction of two words that does not fall into either of the above categories is let's, a contraction of "let us" that is used in forming the imperative mood in the first-person plural (e.g., "Let's go [somewhere]"). Use of the uncontracted "let us" typically carries an entirely different meaning, e.g., "Let us go [free]". "Let us" is rarely seen in the former sense and "let's" is never seen in the latter one. Informal contractions are, by their nature, more frequent in speech than writing, e.g., John'd fix your television if you asked him. Contractions in English are generally not mandatory as in some other languages. It is almost always acceptable to write out (or say) all of the words of a contraction, though native speakers of English may judge a person not using contractions as sounding overly formal. Let's, as mentioned above, is an exception to this rule." http://en.wikipedia.org/wiki/Contraction_(grammar)

Technically，they are different in meaning.When someone says "let's go',he means that he will go with the others.While "let us"probably suggests that the speaker is not included in what they are going to do.

There's no difference. Use whichever you like.

Cannot explain why someone would say that. "Let's" is the contraction of "let us" and all contractions have the identical meaning and usage as their uncontracted forms. They are equivalent.