Probably not an application for just any audience, but I thought I'd share...
The Vandermonde determinant shows up in matrix models of quantum field theories. Roughly speaking, in these we consider integrals of the form
$\int dM f(M)$,
over the space of Hermitian matrices, where $f$ is invariant under conjugation by unitary matrices, and $dM$ is the (also conjugation invariant) measure
$dM = (\prod_i dM_{ii})(\prod_{i\lt j} dM_{ij})$.
We want to calculate the integral by gauge fixing. In other words, to integrate over a judiciously chosen set of representatives of each $U(N)$ orbit. Usually the best set of representatives to take are the diagonal matrices. The procedure for evaluating the integral in that case is exactly that of the Weyl integration formula. In doing so we get what physicists call the Faddeev-Popov determinant, which turns out to be the Vandermonde determinant! In other words, we get an equivalent integral
$\int d\lambda_1 ... d\lambda_N \Delta(\lambda_1, ...,\lambda_N) f(diag(\lambda_1,...,\lambda_N))$,
where $\Delta$ is the Vandermonde determinant.
