I'm not sure this is a well-posed question. Or at least, maybe I misunderstand it. I think what you're interested in is the difference between an analogue and digital PID controller.
If that is the case, the basics aren't so difficult.
An analogue PID controller uses continuous-time, analogue components such as an opamp amplifier, integrator, and differentiator for the P, I, and D respectively. Signals are manipulated in real-time, in hardware, but only at the complexity you designed into the controller. The weight of each component is controlled by a gain parameter; these gains are tuned to get the PID controller to behave as desired. See the Ziegler–Nichols method which just happens to be the method I learned/remember.
Here's the relevant bits of an analogue PID controller pinched from http://www.ecircuitcenter.com/Circuits/op_pid/op_pid.htm
Advantages include: it's fast (lack of any processing delay), and the ability to create simple designs with few/cheap parts (at least for simple controllers.) The analysis and synthesis is typically easier than discrete-time digtal PID controllers.
Disadvantages include: being practically constrained to only a few poles, the fixed nature of hardware is not easily changed or improved later, tuning is a real pain!
A digital PID first samples the inputs using an ADC. After that, everything that is the PID controller is simply mathematics carried out on vectors of samples in a processor, CPLD, or FPGA. The continuous-time calculus breaks down into the discretised finite-differences. The math introduces a processing delay between input and output which you need to design around. Finally a DAC reverses the process to provide the continuous-time output in a process called "zero-order hold."
Here's the block diagram for your linked device.
Advantages include: the ability to realise a high number of poles with no extra effort (at the cost of processing delay), the ability to easily change and improve the design later, easier tuning (software can sometimes even do it for you!)
Disadvantages include: introducing the processing delay which is dependant on the PID design and the processor it's running on, typically more complex/costly than a simple analogue equivalent.