Andy, I'm not a motor engineer, but I can give an overview of the difference.
The ECU has many inputs from sensors on the engine and produces outputs controlling the actuators for the engine controls.
The map relates all of the inputs to a set of outputs
Inputs include accelerator pedal position, fuel temperature, inlet air temperature and pressure, mass air flow, injector lift commencement etc etc.
Outputs include throttle butterfly position (on a petrol car), injection quantity, injection commencement, glow plug power and so on.
The Evry mod falsifies one input parameter to make the fuel quantity higher than the ECU map expects.
This works because the diesel engine almost always has more air than is needed, and burns lean at low power levels.
One consequence of this is that the injection event takes longer to complete. This means that the commencement of injection is later than it should be for the fuel quantity, as the map relies on the falsified signal.
A remap changes the actual map in the ECU. All the sensor signals are unaffected, so the ECU is still receiving correct measurements of what's going on in the engine. The new map sends different signals to the actuators, adjusting all of the controls to deliver a bigger bang at the correct time. Injection timing, quantity, EGR activation, boost pressure, etc. - all are changed in the new map to keep the engine happy.
In simple terms, the standard map keeps the engine's performance well below the point where failure of any components is predicted, even in the case of all the manufacturing tolerances adding up "the wrong way". Remapping pushes performance closer to the edge, and will reduce the life of any components on the edge of their specification. If taken too far, the engine will be overstressed and fail in a more obvious way. Because the engines (especially the lower powered TDI's) are built with a lot of extra capacity, remapping can be taken a long way without seriously affecting the engines reliability.
Having said that, every time you make a modification to the engine's performance, you take a risk that it will cause a failure somewhere. You need to be ready for that to happen, even if it isn't very likely.