A polyhedral complex is a set of polyhedra. Its support is their union.
In other words, a polyhedral complex has an internal structure in terms of what its constituent polyhedra are and how they are arranged. Taking the support "forgets" the internal structure and flattens it into an undifferentiated set of points.
For example, in two dimensions, you can overlay a polyhedral complex over the set $[0,1]\times[0,1]$ in many ways: as a single square-shaped polyhedron; as two isosceles right triangles; as an $n\times n$ grid of squares of side $\frac1n$; and so on. All of these are different polyhedral complexes, but they have the same support.