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.