One minor difference I see is that one of them says: "I want to collect many different proofs." The other one says: "I know some standard approaches, I am interested in less usual proofs." (But I do not think that the answerers bothered about the restriction not to include usual proofs.)
Another minor difference is that in one case it is formulated as $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ and in the other case as $$\left(\frac{x_1+ \ldots + x_n}{n}\right)^n \geq x_1 \cdots x_n.$$
I do not think that either of these two differences is big enough to treat them as two completely different questions.
