As an engineer analyzing a system (whether it be a circuit or an audio sample), you should know when to apply the analysis tools you've been given--such as Discrete Time Fourier Transform and Z-Transform. I understand that a Fourier Transform will tell me how the amplitude of the system changes with respect to inputting different frequencies. For example, I believe that performing the Fourier Transform on a LC circuit would give me information as to what the resonant frequency is. However, I don't know when I should apply the Z-Transform. It's usefulness is not apparent to me. I don't understand what information I would get from the Z-Transform that I wouldn't get from a Fourier Transform.
I realize that the Z-Transform and Fourier transform are very similar, except that the Z-Transform has an 'r' term in it. I don't know the significance of this or of the result coming from a Z-Transform compared to the result coming from a Fourier Transform.
What additional information are we getting by doing the Z-transform, that we are not getting from doing the DTFT?
Edit: It has been suggested that my question is a duplicate of this one: Mathematical advantages of the ZT, DTFT and DT?
My question is different from this one because I am asking from the perspective of an engineer, not from the perspective of a mathematician. I would like to understand the intuition and real world physical meaning behind this. These techniques are all developed for the purpose of solving some real world problem and I want to understand what that is.