Compare Fourier and Laplace transform

Laplace is generalized Fourier transform. It is used to perform the transform analysis of unstable systems. Simply stating, Laplace has more convergence compared to Fourier.

Laplace transform convergence is much less delicate because of it's exponential decaying kernel exp(-st), Re(s)>0. Laplace transform is an analytic function of the complex variable and we can study it with the knowledge of complex variable. Laplace is also only defined for the positive axis of the reals.

you can as well consider $\omega \in \mathbb{C}$ and talk about a "generalized" Fourier tranform, converging in the appropriate domain of $Im(\omega)$, usually a "strip" parallel to the real axis. You may want to check Lukacs 1970, Th. 7.1.1 or Titchmarsh, E.C. (1975): Introduction to the Theory of Fourier Integrals, Oxford University Press. Reprint of the 1948 second edition.