How does spin influence the dynamics of quantum mechanical systems? I have just been introduced to the Klein-Gordon Equation and the Dirac Equation for the first time. The way they were explained to me, these equations govern the (relativistic) evolution of spin-0 and spin-1/2 particles respectively.
My question is, what fundamentally differentiates particles with different spins, and how can those differences have such large effects on the system's dynamics that particles with different spins are described by two fundamentally different equations of motion?
Even though I am still an undergrad, I understand PDEs well enough to know that even slightly altering a system's equation of motion can lead to dramatically different physical solutions. So the fact that spin is able to entirely change the relevant EOMs is quite surprising to me, and seems to suggest that it plays a central role in these systems that I do not yet fully appreciate.
 A: Elementary particles and the composites from them are defined by spin: if spin is a multiple of 1/2 it is a fermion, if it is a whole number it is a boson. This is incorporated in the spin statistics theorem that describes the difference in their wavefunction, leading to the use of different differential equations. The proof is further in the link.
The spin–statistics theorem implies that half-integer–spin particles are subject to the Pauli exclusion principle, while integer-spin particles are not.
As a principle, it comes from experimental observations, and is used axiomatically when choosing the solutions of quantum mechanical equations which describe the data.

The Pauli exclusion principle is part of one of our most basic observations of nature: particles of half-integer spin must have antisymmetric wavefunctions, and particles of integer spin must have symmetric wavefunctions

The reason we needed quantum mechanics was that classical could not fit the data. A basic fact that the exclusion principle is necessary  comes from the very existence of atoms and the periodic table. There would not be the form we see if all electrons could occupy the lowest energy levels, no chemistry ....
You say:

So the fact that spin is able to entirely change the relevant EOMs 

It is the separation between symmetric and antisymmetric wavefunctions that is the drastic "mathematical" assignment  change between bosons and fermions, as discussed in the link.
