What is the advantage of a Z transform derived PID implemenation?
The Z-transform method of designing the PID controller will eventually yield a much more efficient implementation.  This is critical if you are designing for the smallest FPGA/DSP/Microcontroller for you application.

The mentioned "PID without a PHD" is probably the easiest approach to implementing a PID control in software, but it becomes cumbersome at higher sampling rates.

Additionally, the Z-transform lends itself better to designing in a discrete (digital) domain.  The traditional (Laplace transform) method of design is more for continuous time.  There are multiple ways of converting between the two (Zero-Order Hold, Linear Interpolation, Pole Placement, Bilinear/Tustin), each comes with it's strengths and drawbacks in terms of stability and response of the system.  It's generally just easier to do the entire design in the discrete domain.

Long story short, if you are using a relatively "slow" system (all of the major behaviors happen significantly under a 100kHz or so), then the first design is probably just fine.  You can implement it on a microcontroller or PC and be done with it.  As the systems get faster, then you may have to use the Z-transform method to get the speeds that you need (the article mentions 9.5MHz, assuming you have A/D and DACs that can keep up).