Regarding Strain Energy & Strain Energy (Density) here is some more info.
The units of energy are force*distance, so when a load is applied and the material deforms, we are putting energy into the material. This energy introduced into the material due to the loading is referred to as “strain energy.” We prefer to normalize strain energy by unit volume, and when we do so, this is referred to as strain energy density. The area under a stress-strain curve is the energy per unit volume (stress*strain has units of force per area such as N/mm2 , which is the same as energy per unit volume N-mm/mm3 . We will be assuming linear elastic material only. Most metals and alloys are linear elastic prior to the onset of plastic deformation, so this is a valid assumption.
Though Element Strain Energy is a good measure to watch in Modal analysis, as it gives an idea on the area of concern & to go for the mesh refinement there as ESE in modal analysis is somewhat similar to what Stress in Static analysis.
But if we go for Static/Structural analysis then Strain Energy Density could give you an idea on what elements/area of your model are contributing more in taking the load (absorbing). This same factor is used in optimization as well, where the load-paths are identified where the model have high strain energy density. So basically you get idea where you should strengthen your part. High stresses & High strain energy could be on common areas in some cases but not always guaranteed.
So better to go for Stress results when your main concern is Stress (which is there in most of the cases). But when your main concern is Stiffness of the structure i.e you want to limit the deflection/displacement happening in your structure then you may want to stiffen your part wherever it is showing high strain energy density.
Just to add, Strain Energy Density dependent function is also the main criteria on which Hyperplastic materials are based, as the behavior of Hyperelastic materials can’t be described with a single modulus like we describe for a general elastic-plastic material.
Refer below link for ESE.