The difference is that some of Maxwell's early formulations focused more on the scalar and vector potentials, while Heaviside's formulation focuses on the electric and magnetic fields. The potentials are more natural for quantum physics, and are arguably physical entities observable in quantum effects (Aharonov–Bohm effect), but in classical electrodynamics both formulations are exactly equivalent. Heaviside's formulation doesn't leave out any interesting physics present in Maxwell's original formulation, though there is interesting physics not predicted by classical electrodynamics.

> essentially the Heaviside equations are designed to be easy for electrical engineers to use, through an appropriate choice of gauge (Lorenz), which limits their true scope

As for your linked paper - it is full of inaccuracies and faux-mysteries that are actually either well explained in mainstream physics or simply manufactured by the author. I'll respond to some of the more blatant problems below.

> Then there is one ofour first elephants in the room. If this simplification of reformulated 12 of 20 equations to give us todays mathematical framework, what happened to the other eight equations?

Here's what happened to them: they have been lost, rejected or forgotten by mainstream physicists.

Just kidding! Nowadays we call them "Ohm's Law," "conservation of charge", and the definition of the electric displacement field. This is not a secret - you can read Maxwell's 1865 paper and compare to the modern version, or it's even on your wikipedia page.

> [ various whining about the displacement current ]

Please refer to any source on electrodynamics. It is widely acknowledged that "displacement current" is not really a current, it's just a name for the contribution of a time-varying electric field to magnetic fields.

> The constant mu0 is a constant that is measured from the forces created from electric currents flowing thru COPPER conductors.These relations have worked fine for electric currents flowing thru COPPER wire conductors. However,when we use this constant for other types of electric currents in different substances we end up with the wrong values and the wrong assumptionson how these materials react to electric currents

The permeability of free space is not dependent on copper conductors.

> Yet,while in the software worldleaving loose ends like uninitialized variables or unconnected interrupt vectors is considered sloppy at best. The physics world seems to be very happy with leaving loose ends,like extra equations and unsolved potentials,around with no real effort to resolve them.

Yeah, gauge freedom is a thing. It's fine.

A lot of this "paper" seems to be freely mixing the original formulations of Maxwell's equations with some "generalization" of Maxwell's equations proposed in this:

http://www.andre-waser.ch/Publications/GeneralisationOfClass...

(cited on page 39), which adds some random terms the equations to yield a new physical scalar field, which no one has ever seen.

I'd strongly recommend that you find a better source for your study of electrodynamics than this document by some dude trying to sell yet another propellantless vacuum thruster.