What is the difference in working of an GMR readhead vs an EMR one?
This drawing shows a better overview of the reading head: 

Here you can see that my assumption that they both operated with a stack was completely wrong. The diagrams above where in fact the reading head itself and represent an alternative to the stacked layout. This can be seen by the fact that the Au shunt is in magnetic proximity to the magnetic grain.

So the difference between GMR en EMR is the entire structure of the reading head and underlying physics. They provide the same function (reading of magnetic grains) but GMR uses an alternating series of magnetic and non-magnetic layers to read out while EMR uses a Au shunt resistance separated from the 4-point measurement pins by a InSb barrier.

As @Andy_aka pointed out, the working physics behind this setup depends on knowledge of the materials involved and is perhaps better suited for a physics forum. If the community feels differently about this I will append my answer.