On the simplest end of that spectrum, Taylor series are useful because many real-world dynamics can be approximated as a "primarily linear behavior" + "nonlinear effects."

(And cases where that isn't true can still be instructive - a Taylor series expansion for air resistance gives a linear term representing the viscosity of the air and a quadratic term representing displacement of volumes of air. For ordinary air the linear component will have a small coefficient compared to the quadratic component.)