What are the advantages of Laplace Transform vs Fourier Transform in signal theory?
Laplace transforms can capture the transient behaviors of systems.  Fourier transforms only capture the steady state behavior.  Of course, Laplace transforms also require you to think in complex frequency spaces, which can be a bit awkward, and operate using algebraic formula rather than simply numbers.

If you want to see the power distribution of a signal over time, Fourier transformations are often the easiest way to do it.  However, if you want to understand what a system does when you flip a light switch, you typically need Laplace transforms.