Connection with system analysis and laplace&Z transform
You are right that the (bilateral) Laplace transform can be interpreted as the Fourier transform of $e^{-\sigma t}f(t)$. However, I think that the significance of the Laplace transform only becomes clear when $s=\sigma+j\omega$ is viewed as a complex variable because then we can study the analytic properties of the system function. E.g., electrical networks are characterized by the poles and zeros (in the complex $s$-plane) of their system function.

The practical importance of the Laplace transform for the analysis of continuous systems lies in the fact that it transforms linear differential equations with constant coefficients into algebraic equations, which are obviously much easier to solve. This is achieved by the fact that in the Laplace transform domain, integration is equivalent to multiplication by $1/s$, and differentiation becomes multiplication with $s$.

Any linear time-invariant (LTI) system, such as the RLC-circuit you mentioned, will result in linear differential equations with constant coefficients. This is why any LTI system can be analyzed by the Laplace transform. Note that in practice usually the unilateral transform is used (where the integral starts at $t=0$ instead of $t=-\infty$), because practical systems are causal, and because in this way initial conditions (such as charged capacitors at $t=0$) can be easily taken into account. The unilateral Laplace transform is thus an appropriate tool to solve initial value problems.

Everything said so far carries over to the $\mathcal{Z}$-transform if you replace 'continuous system' by 'discrete system', and 'differential equation' by 'difference equation'. Linear time-invariant discrete systems consisting of adders, multipliers and delay elements are treated by the $\mathcal{Z}$-transform in the same way as LTI electrical networks are treated by the Laplace transform. And indeed, IIR filters are analogous to RLC circuits in the sense that both have an infinitely long impulse response and that both can be characterized by the poles and zeros of their system functions.