A garden arch is usually used to frame the entrance to a garden, or to reach over a garden path. Arches are usually made of wood, and can support most climbing and twining vines.
Trellises are more varied than arches. Though usually a tripod-like structure with ladder-like size, trellises come in many shapes and sizes, and can be made of wood, wire or even twine.
Trellises and arches are most useful in areas where you'd like to grow vines, but don't have a structure to support them. Plant trellises near fences or walls, as opposed to the middle of a lawn. In a garden, use a trellis to support beans, peas and smaller melons.