(1) Not really an application, but the paper by I. Gessel, Tournaments and Vandermonde's determinant, J. Graph Theory 3 (1979), 305-308, gives a nice connection with tournaments. See also Exercise 2.16 of my book Enumerative Combinatorics, vol. 1 (equivalent to Exercise 2.35 at http://math.mit.edu/~rstan/ec/ec1.pdf).
(2) Probably not suitable for an undergraduate course, but if $s_{(n-1,n-2,\dots,1)}$ denotes the Schur function of the staircase shape $(n-1,n-2,\dots,1)$, then the evaluation $$ s_{(n-1,n-2,\dots,1)}(x_1,\dots,x_n)=\prod_{1\leq i \lt j\leq n} (x_i+x_j) $$ follows immediately from the bialternant formula for Schur functions, since it reduces to the quotient of two Vandermonde's: $\prod (x_i^2-x_j^2)/\prod(x_i-x_j)$. See Exercise 7.30 of Enumerative Combinatorics, vol. 2.
