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.
In addition, the Fourier Transform exists along the $j \omega$ axis of the two sided Laplace Transform so Fourier is a slice of Laplace.
The $\sigma$ axis captures the dissipation/regeneration properties of Linear Systems. While there are Bode criteria for stability in the frequency transfer function, the $\sigma$ axis is very useful in feedback stability analysis like a root locus plot.
The Fourier is used primarily to describe signals while the Laplace is associated with the system that the signal propagates through.
These concepts generalize naturally to higher dimensions such as complex wave number. A $\sigma$ can Model things like skin effects in wave guides.
Broadly speaking (and over-simplifying), I would say that Laplace Transform (LT) is better-suited for symbolic computation and systems analysis whereas Fourier Transform (FT) is very useful for numerical computation and signal analysis.
One notable differences is the mathematical space on which they operate: FT is well-defined on $L_2$ space where we have Plancherel Theorem (and even on $L_1$ space), whereas the LT is (generally) defined only for functions of $t>0$ (or sometimes, both of them called monolateral LT).