Garrett Birkhoff used the Hilbert metric (he called it the projective metric) to prove that every $n$-by-$n$ matrix $A$ with positive entries is a Hilbert metric contraction on the cone of nonnegative vectors. He gave a formula for the contraction constant which is $\frac{1}{4} \arctan \Delta$ where $\Delta$ is the maximum Hilbert's (projective) metric distance between $Ae_i$ and $Ae_j$ where $e_i$ and $e_j$ are distinct elementary basis vectors. This immediately implies one aspect of the Perron-Frobenius theorem: that matrices with positive entries have a unique Perron eigenvector.
The Hilbert metric proof of the Perron-Frobenius theorem also extends to nonlinear maps which are monotone and homogeneous of degree one (i.e., $f(\lambda x) = \lambda f(x)$, $\forall \lambda > 0$). This is the primary advantage of using the Hilbert metric to prove the Perron-Frobenius theorem.