I just found this post will looking, into exactly what the difference is in these two triggers, it turns out the answer is actually in the difference between ATriggerBase and ABrush.
ATrigerBase, provides a UShapeComponent (https://docs.unrealengine.com/en-US/API/Runtime/Engine/Components/UShapeComponent/index.html) to represent a collision volume, thus you can have the 3 basic collision primitives of Sphere, Capsule and Box used by Unreal, hence these are going to be the most efficient possible (better than using a hidden StaticMesh as a Collision Volume).
ABrush, on the other hand, uses UBrushComponent (https://docs.unrealengine.com/en-US/API/Runtime/Engine/Components/UBrushComponent/index.html) and UBrushBuilder (https://docs.unrealengine.com/en-US/API/Runtime/Engine/Engine/UBrushBuilder/index.html), This provides the basics of the Geometry Brush System (i.e. BSPs). Based on what the documentation says for these it appears that for ATriggerVolume, a set of Convex Meshes will be procedurally generated to represent the Collision Volume. So I expected the cost ends up the same as using BSPs. In theory, depending on exactly how the Brushes are being implemented, Using a Box brush may end up equivalent to indirect use of a Box Shape, but there is nothing in the documentation, that I can find to suggest that this is actually the case.
Base on this I believe it’s generally better to use the Trigger regions, derived from ATriggerBase, where possible.