1. Find the electronic configuration of $\mathrm{Fe^{3+}}$

Fe has atomic number $26$.

$\mathrm{Fe}: [Ar]3d^6 4s^2$

For $\mathrm{Fe^{3+}}$, remove two $4s$ electrons and one $3d$ electron:

$\mathrm{Fe^{3+}}: [Ar]3d^5$

So, $\mathrm{Fe^{3+}}$ is a $d^5$ ion.

2. Case 1: SCN$^-$ (thiocyanato-S) ligand environment

SCN$^-$ coordinated through sulfur is a weak field ligand. In an octahedral field, a weak field ligand gives a high-spin configuration.

For octahedral high-spin $d^5$:

$t_{2g}^3 e_g^2$

Number of unpaired electrons:

$n = 5$

Spin-only magnetic moment:

$\mu = \sqrt{n(n+2)} = \sqrt{5(5+2)} = \sqrt{35} \approx 5.92\ \text{BM}$

Approximated to nearest integer:

$\mu \approx 6\ \text{BM}$

3. Case 2: CN$^-$ ligand environment

CN$^-$ is a strong field ligand. In an octahedral field, it gives a low-spin configuration.

For octahedral low-spin $d^5$:

$t_{2g}^5 e_g^0$

Number of unpaired electrons:

$n = 1$

Spin-only magnetic moment:

$\mu = \sqrt{1(1+2)} = \sqrt{3} \approx 1.73\ \text{BM}$

Approximated to nearest integer:

$\mu \approx 2\ \text{BM}$

4. Find the required difference

$6 - 2 = 4$

So, the difference between the spin-only magnetic moments is

$\boxed{4}$