Visualising a Z-transformed Transfer Function?
in continuous-time functions, like $x(t)$, the operation of the derivative makes sense, it is well defined.  the three components to a continuous-time LTI filter are adders (two or more signals being added), scalers (a signal is simply multiplied by a constant), and integrators (the $s^{-1}$ operators).  the first two components do not discriminate with regard to frequency, so with just those two components, you cannot make a "filter" that filters out some frequencies more so than others.  but the integrator does act differently on sinusoids of different frequencies.  higher frequencies come out of the integrator reduced in amplitude more than lower frequencies.

in discrete-time functions (or "sequences"), like $x[n]$, the derivative operator does not makes sense, it is not defined.  the three components to a discrete-time LTI filter are adders (two or more signals being added), scalers (a signal is simply multiplied by a constant), and delay elements (the $z^{-1}$ operators).  the first two components do not discriminate with regard to frequency, so with just those two components, you cannot make a "filter" that filters out some frequencies more so than others.  but the delay element does act differently on sinusoids of different frequencies.  higher frequencies come out of the delay element shifted more in phase than lower frequencies.

analog filters act on physical quantities in time.  digital filters act on numbers.  these numbers are samples of a continuous-time function ($x[n]=x(nT)$) and make up a sequence.  there is no way to do derivatives or integrals directly.  but we can delay any of these sequences by use of computer memory.  that's why we talk of $H(z)$ more so than $H(s)$ when designing and implementing digital filters.