I would love to get professional answers on this in general. In the meantime here's my crude attempt using the Simple Harmonic Oscillator as an example..
Consider a function of an integer variable defined by this..
where $n=2,3,4,5,..$ and $k$ is a constant.
If $k$, $f(0)$ and $f(1)$ are given then $f(n)$ can be calculated for any $n$.
This equation is an example of a difference equation. An area of mathematics called the Theory of Finite Differences (or Difference Calculus) tells how to solve these equations. The solution is a nice surprise..
It's the famous sine function, where $a$ is a constant related to $k$, $k=2*cos(a)$. So the solution is a wave and if you plot $f(n)$ for $n=2,3,4,5,...$ you get the beautiful sine wave, and the three numbers $k$, $f(0)$ and $f(1)$ determine the amplitude, wavelength and phase of the wave.
What about $n$? It plays the role of time, because at time=$n$ the function $f(n)$ is the displacement from the origin for a Simple Harmonic Oscillator.
So the difference equation $f(n)=k*f(n-1)-f(n-2)$ replaces the second order differential equation used to describe the Simple Harmonic Oscillator. It's a nice example of Difference Calculus in action.
Of course, things are not exactly the same.. time is no longer a continuous variable!