They Did It Together

When we superimpose both the effects of change in speed along the direction of incidence, that gives us Snell's Law and boundary conditions applied to the $B$ field at the medium interface, we have,

$\alpha\to\alpha_2$

For the case of $\mu_2\gt\mu_1$,

$\alpha_{2s}\lt\alpha_2+\Delta\theta$

and

$\alpha_{2p}\gt\alpha_2-\Delta\theta$

For the case of $\mu_2\lt\mu_1$,

$\alpha_{2s}\gt\alpha_2+\Delta\theta$

and

$\alpha_{2p}\lt\alpha_2-\Delta\theta$

When $\Delta \theta=0$, the equations collapse to

$\alpha_{2s}=\alpha_{2p}=\alpha_2$

as we observe by Snell's Law for refraction alone.  (We take the intersection of the two regions defining $\alpha_{2s}$ and $\alpha_{2p}$ after admitting the case of $\mu_1=\mu_2$. )

It might appear that the split is due to the change in polarization $\theta$ alone.  In fact, both polarization, $\theta$ and boundary conditions for the $B$ field at the interface, contribute to refraction.

Note: $\mu\ne n$