Coil Area vs Core Area
How is the difference in inductance related to the core to coil area
  ratio?


It's a good question but there will be "nuances" that means this answer is not 100% correct for all situations. Start with magnetic reluctance \$\mathcal{R}\$ and apologies if the math goes round the hills a couple of times. 

It is defined thus: -

$$\mathcal{R} = \dfrac{\ell}{\mu\cdot A}$$

Reluctance is the length of the core divided by the permeability x the cross sectional area. Reluctance is also (more traditionally) defined as: -

$$\mathcal{R} = \dfrac{N\cdot I}{\Phi}$$

Here, reluctance is the number of turns (N) multipled by the ratio of applied amps to the magnetic flux produced. This basically tells us that a higher reluctance produces less flux  per amp. It's probably what most folk are used to when understanding reluctance.

If these two formulas are equated we get: -

$$\Phi = \dfrac{\mu\cdot A\cdot I\cdot N}{\ell}$$

If we differentiate flux w.r.t time we get: -

$$\dfrac{d\Phi}{dt}= \dfrac{\mu\cdot A\cdot N}{\ell}\cdot \dfrac{di}{dt}$$


We can use Faraday's induction law to equate V/L to \$\frac{di}{dt}\$
And We can equate V/N to \$\frac{d\Phi}{dt}\$
V is voltage, L is inductance


We now get the well-known formula for inductance: -

$$L = \dfrac{\mu\cdot A\cdot N^2}{\ell}$$

From the top we can substitute \$\ell\$, \$\mu\$ and \$A\$ for reluctance and we get: -

$$L = \dfrac{N^2}{\mathcal{R}}$$

Note that this formula is a slightly rearranged version of \$A_L\$, (core inductance factor) seen in ferrite data sheets with \$A_L\$ being the inverse of reluctance (permeance).

We can "estimate" the reluctance of the air between the ferrite core and the coils by calculating the area it occupies in the overall cross-section of the coil then applying it into the formula right at the top. 

Then, noting that reluctances in parallel sum-together like resistors in parallel, we should be able get a composite value for reluctance comprising air and core material.

Use this composite value in the bottom formula and bingo. 

Where this method needs work (and where my understanding lets me down) is in "estimating" the reluctance of the air within the coil's cross-section - it may not be as simple as calculating the overall area it occupies because there may be nuances about the air-shape that means it's not generally applicable.