Certain contractions set a tone. Let's gives a more youthful energy to me, but Let us gives off an air of sophistication and elegance.
Here, I'll try to make an example.
"Let us fly on the feathered wings of kings
Let us reach out and touch the starry heaven
Let us see, eyes open for what life brings
Let me see, once gone, my life's impression."
Compare that against this.
"Let's run in the fields and over the hills
Let's Roam in woods to find our treasure,
Let's turn every stone to find all the thrills
Let's gain from life every single pleasure."
Not the best, but I hope you see what I'm talking about. 'Let us' would not have felt right in the second poem, instead, in my opinion at any rate, 'let's give the piece a nice touch, whereas in the first poem 'let us' is an integral part of setting the tone.