One context in which this relationship between the hypercube and the permutahedron appears is as follows. The space of weights of the Lie algebra $\mathfrak{sl}_n$ is naturally identified with $\mathbb R^n/(1,\dots,1)$. Under this identification vertices of the unit hypercube (other than $(0,\dots0)$ and $(1,\dots,1)$) project precisely into all Weyl group translates of fundamental weights. These are the primitive vectors of the braid fan (Weyl fan) and their convex hull is dual to the permutahedron. In other words, the orthogonal projection along $(1,\dots,1)$ takes the hypercube to a dual of the permutahedron and identifies the subcomplex in Richard's answer with the boundary of said dual polytope. It also provides a bijection between the $n!$ maximal simplices of the mentioned triangulation and the $n!$ Weyl chambers.
This can be generalized by replacing the hypercube by an arbitrary order polytope and the permutahedron by its sum with a certain cone.