Does the Laplace Transform have any practical use or provide any mathematical insight? 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?

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.
