The signal model for Laplace is x(t)*exp(alpha t) where as the signal model for HT is x(t) only. Another way to say it is that Laplace models the signal envelo

The signal model for Laplace is x(t)*exp(alpha t) where as the signal model for HT is x(t) only. Another way to say it is that Laplace models the signal envelope as an exponential explicitly while decomposing the remaining signal into frequency components. In Fourier space, the exponential is "rolled into" the frequency components.