Why does a loop antenna not measure the differential magnetic field?
Electromagnetic waves, when they are far from the source, and far from electrically conductive or ferromagnetic materials, have electric field and magnetic field components that are related to one another by the relationship
$$E = \sqrt{\frac{\mu}{\epsilon}}H$$
It is only in "near fields" that the \$E\$ and \$H\$ components can vary independently.
So, when one is measuring a "far field" it doesn't matter whether one measures the \$E\$ field, or the \$H\$ field, or some combination of the two.
When one is measuring a "near field", with a loop antenna, the type of loop antenna makes a difference.
There are "small" loop antennas and "large" loop antennas. The "large" variety have a loop perimeter that is approximately one wavelength of the frequency of interest. The "small" variety have a loop perimeter that is less than one half of a wavelength.
The "large" loop antennas have peak radiation (or reception) directions that are perpendicular to the plane of the loop. The peak radiation (or reception) directions of a "small" loop antenna are along the plane of the loop.
Now, importantly for your question, when a small loop antenna is receiving, it is driven more by the magnetic field than by the electric field, whereas a large loop antenna is driven by both electric and magnetic fields. A small loop antenna with a ferromagnetic core, known as a "loop stick", is almost exclusively driven by the magnetic field.
So, the answer to your question
Why does a loop antenna not measure the differential magnetic field?
The answer is, it can, if it is the right sort of loop antenna (i.e. "small" in relation to the wavelength). It will especially do so if it is a loop-stick antenna.
I am making the assumption that a loop antenna is essentially an inductive loop and behaves as such, please let me know if this is incorrect.
That is not correct for large loops. The get part of their energy from the H field, and part from the E field.
Knowing that the voltage (emf) across an inductor is proportional to the rate of change of magnetic field, why when we measure the voltage from a magnetic loop antenna are we not measuring the rate of change in time of the magnetic field rather than the magnetic field magnitude directly?
When you measure the voltage at the feedpoint of a small loop antenna, you are measuring the rate of change of the magnetic field, rather than its instantaneous value.
I am trying to understand the difference between conventional loop antennas and B-dots (magnetic differential field probes) but the physics seems to be the same.
Magnetic differential field probes are very "small" (relative to wavelength) loop antennas. So the effect of the electric field on them is "small" as well.
