How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

This can also be done quite simply using characteristic functions.

The characteristic function for the Gumbel($\mu$,1) is $\Phi(t)=e^{i\mu t}\Gamma(1-it)$.

For independent variables X and Y, their linear combination has characteristic function:

\begin{align} \Phi_{X-Y}(t)&=\Phi_{X}(t)\Phi_{Y}(-t)\\ &=e^{i\mu_X t}\Gamma(1-it)e^{-i\mu_Y t}\Gamma(1+it)\\ &=e^{(\mu_X-\mu_Y) it}\Gamma(1-it)it\Gamma(it) \end{align} where the last line uses the fact that the Gamma function satisfies the functional equation $\Gamma(1+z)=z\Gamma(z)$.

We then use Euler's Reflection Formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ to get

\begin{align} \Phi_{X-Y}(t)&=e^{(\mu_X-\mu_Y) it} i t\frac{\pi}{\sin(\pi it)}\\ &=e^{(\mu_X-\mu_Y) it} \frac{\pi t}{-i \sin(\pi it)}\\ \end{align}

which is the characteristic function for the logistic distribution.

The change of variables $u=\mathrm e^{-w}$ transforms this into a neat gamma integral, to which one can apply the change of variable $v=(\mathrm e^z-1)u$ to conclude.

Note that the transformation $w\to\mathrm e^{-w}$ is ubiquitous in quite a few Gumbel related manipulations.