It is challenging to identify metrics that best capture hurricane destructive potential and costs. Although it has been found that the sea surface temperature and vertical wind shear can both make considerable changes to the hurricane destructive potential metrics, it is still unknown which plays a more important role. Here we present a new method to reconstruct the historical wind structure of hurricanes that allows us, for the first time, to calculate the correlation of damage with integrated power dissipation and integrated kinetic energy of all hurricanes at landfall since 1988. We find that those metrics, which include the horizontal wind structure, rather than just maximum intensity, are much better correlated with the hurricane cost. The vertical wind shear over the main development region of hurricanes plays a more dominant role than the sea surface temperature in controlling these metrics and therefore also ultimately the cost of hurricanes.