Pulsar wind nebulae are a prominent class of very high energy (E > 0.1 TeV) Galactic sources. Their Gamma-ray spectra are interpreted as due to inverse Compton scattering of ultrarelativistic electrons on the ambient photons, whereas the X-ray spectra are due to synchrotron emission. We investigate the relation between the Gamma- and-X-ray emission and the pulsars' spin-down luminosity and characteristic age. We find that the distance-independent Gamma- to X-ray flux ratio of the nebulae is inversely proportional to the spin-down luminosity, (\propto \dot{E}^-1.9), while it appears proportional to the characteristic age, (\propto tau_c^2.2), of the parent pulsar. We interpret these results as due to the evolution of the electron energy distribution and the nebular dynamics, supporting the idea of so-called relic pulsar wind nebulae. These empirical relations provide a new tool to classify unidentified diffuse Gamma-ray sources and to estimate the spin-down luminosity and characteristic age of rotation powered pulsars with no detected pulsation from the X- and Gamma-ray properties of the associated pulsar wind nebulae. We apply these relations to predict the spin-down luminosity and characteristic age of four (so far unpulsing) candidate pulsars associated to wind nebulae.