Basic difference between Fourier transform and laplace transform?
Fourier transform is an intuitive tool that's a bridge between domain of physics and mathematics, as it quantitatively describes the periodic content of the signals and also frequency response characterisation of systems that occur in physical (and engineering) applications. The use of frequencies is quite intuitive and consistent at least for stable systems...

However for unstable systems (and signals) Fourier transforms becomes mostly awkward (if not useless) to deal with. However control engineers unavoidably must make frequent use of unstable systems in their works. And for this purpuse, Fourier transform is either insufficient or awkward, hence a generalisation of the existing Fourier transform is made into the Laplace transform which conveniently yields mathematical (complex algebric) descriptions of stable as well as unstable systems which was not possible with the Fourier. The Laplace transform, therefore, includes a region of convergence parameter into it.

Another difference between the two transforms is in the time-domain transient analysis of output of LTI systems driven under nonzero initial conditions which is successfully captured in the Laplace transform only. In the sense that LCCDE with initial conditions are straightforwardly solvable by (unilateral) Laplace transforms whereas the standard FT can only solve LCCDE with zero initial conditions (initial rest)...

For one sided and two sided differences, I think Stanley has things to say...