When is it correct to use $\Gamma(n)$ instead of $n!$ when $n$ is a positive integer?

Question

I have seen the notation $\Gamma(n)$ employed in many articles in which $n$ is defined as a nonzero positive integer, instead of $n!$. What is the advantage of using $\Gamma$ in such a context, given that it is the continuous analogue of the factorial?

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EDIT

I am aware that $\Gamma(n)=(n-1)!$. My question relates more to the "concept" of using $\Gamma$ instead of the factorial function.