Is the Laplace transform essentially a generalized version of the Fourier transform? As a beginner, my understanding of Fourier and Laplace transformations is limited. However, it appears to me that the Fourier decomposes functions into a combination of waves, while the Laplace does the same thing except using exponentials.
Additionally, I also often hear that the Laplace can be applied to a broader range of functions than the Fourier, though the reason for this is unclear to me.
I would appreciate if someone could explain the current relationship between Fourier and Laplace transforms and give a short summary comparison.

*

*What are the differences/similarities between Laplace & Fourier in what they accomplish?


*What does the Laplace do that the Fourier does not which makes it apply to more functions?
If possible, I would like the explanation to be in simple language and without relying heavily on advanced mathematical jargon. Thank you.

A Google request like "differences Fourier Laplace" will provide you many many answers.
I react, very politely, to what @user1952009 has said. For me, and I am not alone, "Fourier Transform" and "Laplace Transform" are very different not only on a mathematical point of view, but also for practical needs.
Fourier Transform belongs to the huge "realm" of harmonic analysis, with a far-reaching extension in group theory.
Whereas Laplace Transform belongs to the domain of complex function theory with abscissas of convergence., etc., that have no equivalent in Fourier Transform theory and vice versa. See (materia.fisica.unipd.it/salasnich/dfl/dfl.pdf)
What is misleading is the similarity of computation rules. But, even on an engineering point of view, one should not say "Fourier = Laplace with $s$ replaced by $2 i \pi \nu$ (or $ i t$)". There are too many differences. One big difference among others: with Laplace, convolutions become products, but not in the other way, whereas, for Fourier, it works in both directions.
