"Let's" vs. "lets": which is correct?

Say I'm promoting a product. Which is correct?

• [Product] let's you [do something awesome].
• [Product] lets you [do something awesome].

Or neither?

Let’s is the English cohortative word, meaning “let us” in an exhortation of the group including the speaker to do something.

Lets is the third person singular present tense form of the verb let meaning to permit or allow.

In the questioner’s examples, the sentence means to say “Product (allows/permits you to) do something awesome”, so the form with lets is correct.

The second one is correct.

Let's is a contraction of let us. "Let's go to the ballgame today!"

Lets is the third-person singular simple present indicative form of let -- but of course we all knew that already....
It means allows, which looks like what you want to convey here.