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Mathematics essentially is the study of the transformation of symbols according to specific rules (inference rules/axioms). However we don't just study any arbitrary system of mathematics; we carefully pick ones that make it easier to understand and create solutions for the real world.

What exactly has the Laplace Transform helped simplify or understand better (in addition to the various simpler tools we already have, Fourier Transforms and Linear Algebra)?

In control systems analysis, I've seen the Laplace transform being used to create $s$-domain transfer functions that represent linear time-invariant systems. However, when various graphs for analysis are created (Nyquist, Nichols, Bode) we substitute $\mathbb{j}\omega$ into the $s$-domain functions. This would make it equivalent to a Fourier transform.

The poles of the $s$-domain function are used to help understand the stability of a system. It can be shown that these poles are the same as the eigenvalues of the linear operator; however the eigenvalues of the linear operator can be clearly associated with their time-domain modes and gives a direct understanding of why an LTI-system is stable if the eigenvalues are all less than 0.

Is there any particular application of the Laplace transform that clearly makes something easier to understand or do? Why is the Laplace transform so heavily taught in schools?

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    $\begingroup$ The Laplace transform is defined for a larger class of functions than the Fourier transform, albeit there are many similarities. $\endgroup$
    – copper.hat
    Commented Oct 29, 2016 at 5:54
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    $\begingroup$ I can think of many. For example, Analysis of electrical circuits, linear ODEs, etc. Note that in control systems, if you work with Fourier transform instead of Laplace transform, the Fourier transform of the unit step function would have an additional $\delta(\omega)$ and you would get into a lot of trouble $\endgroup$
    – polfosol
    Commented Oct 29, 2016 at 5:58
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    $\begingroup$ Thanks polfosol, copper.hat. I do understand that Laplace transforms apply to a larger class of functions because it can exponentially damp them out. But what exactly does the Laplace Transform simplify in the analysis of electrical circuits or linear ODEs that cannot be done already with the use of Fourier transforms or Linear Algebra? Very large LTI systems can be solved with ease with computer linear algebra. Fourier Transforms can be used to create efficient algorithms for convolution. What has the Laplace Transform brought us? $\endgroup$
    – Pradu
    Commented Oct 29, 2016 at 6:07
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    $\begingroup$ If we can dig a hole by a spoon, then why bother using a shovel? $\endgroup$
    – polfosol
    Commented Oct 29, 2016 at 6:14
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    $\begingroup$ Heuristically: (one-sided) Laplace transforms are well-suited for initial value problems; Fourier transforms are well-suited for boundary value problems. I think the reason you can't see the difference is that you aren't thinking precisely enough about particular problems. $\endgroup$
    – symplectomorphic
    Commented Oct 29, 2016 at 6:24

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In all applications of Laplace transform the transformed or backtransformed objects are given or unknown functions realized in terms of finite expressions. The sought after end result should again be a certain expression, found by means of applying the known rules and lookups in a large catalogue.

Nobody would Laplace transform a data set (time series) and look at the resulting graph in order to better understand the underlying physical process. This is in sharp contrast to Fourier transform. The latter is not only applied to finite expressions as an elegant means to come up with nice "analytical solutions" to specific problems with a lot of inherent symmetries, but it is as well applied to one- or two-dimensional data sets in order to detect hidden structures in these data, and for other purposes.

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