Higher order Elements allow you to represent the "fields" with equally accuracy with a coarse mesh with less computational effort. Each higher order element has certainly more unknowns then a first order element, but since you can use a coarser mesh the total effort is reduced.
The problem is: At some parts in your structure the mesh might already be very dense just because of the resolved geometry. Think about the fields inside a plate capacitor. In between the plates, the field is pretty much homogeneous and can be represent almost perfectly with single order elements. Using higher order elements at this location would just increase the computational effort without giving you any extra accuracy.
F.