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Both are used to transform signals and computing algorithms from the time domain (V=f(t)) into another domains (V=f(s) or V=f(z)) to make certain computation easier.
A common application is the design of filters. Instread of dealing with integrals and differentials, filters can be comparatively easily designed in the "s" domain (Laplace transform) where multiplication and division can be used due to the properties of the transformation. Often a Fourier transform is used instead of a Laplace transform.
The "z" transform is afaik mainly used in the design of digital filters where you deal with discrete (digital) signals, not continuous (analog) signals.
To learn more about these transforms and how to use them you need to read some training material (find it on the internet using your favorite search engine).