In line with Joel's answer and the theme that stronger set theories permit finer analysis of higher infinities, an example from commutative algebra suggesting the desirability of distinguishing more than three broad classes of cardinals is (due in final form to Eda):
$Hom(\mathbb{Z}^\kappa / \mathbb{Z}^{<\omega}, \mathbb{Z}) \neq \lbrace 0 \rbrace$ if and only if there exists an $\omega_1$-complete non-principal ultrafilter on $\kappa$.
Whether the class $\lbrace \kappa : Hom(\mathbb{Z}^\kappa / \mathbb{Z}^{<\omega}, \mathbb{Z}) \neq \lbrace 0 \rbrace \rbrace$ is non-empty will depend on whether there are measurable cardinals.
An example from measure theory relates to Fubini's theorem, a central result of core mathematics. In 1990, Joseph Shipman proved (following results of Harvey Friedman on Tonelli-type theorems) that strong versions are provable once real-valued measurable cardinals are around:
J. Shipman, Cardinal conditions for strong Fubini theorems, TAMS 1990