Control systems and convolution
This's a too broad question including continuous time systems, discrete time systems, control systems, convolution operation, difference operation and differentiaiton operations.

However your main focus is on convolution. Therefore I will state in plain simple terms that the operation of convolution defines and calculates the output of a linear time invariant system to a given input from the impulse response of the system. This is performed by the convolution integral for continuous time systems and convolution sum for discrete time systems. Therefore convolution forms the core of signal processing. Just as integration or differentiation forms the core of calculus.

Control theory is an application of extensive signal processing, feedback theory, linear algebra, calculus, matrix theory, transform theory and probability theory. Therefore it will naturaly utilize convolutions in its many formulations. Nevertheless convolution is nothing more than a tool for the control field point of view.

A physical process requires an analog (continuous time) system to control it. But modern technology enables digital computers (sampled data systems) to be used instead. As a consequence, integrations and differentiations of the continuous time system are replaced by sums and differences in digital software implementation of analog systems. Whereas the link between the digital and continuous time systems can be defined via sampling theorem or some other mapping relations.