Naive formulations of QFT are often ill-defined to begin with, and this shows up in calculations as UV divergences. This problem would go away if we started with a theory that is mathematically well-defined. In many cases we can do this by replacing space with a discrete lattice (which eliminates UV divergences), but that's obviously artificial. In some cases we can do it without such contrivances, but a general understanding of how to do it properly is still somewhere in the future.
Pauli-Villars is an approach to fixing this within the context of perturbation theory. It is a perturbative regulator. What does it mean physically? Nothing, really. On the contrary, like the non-perturbative lattice regulator, it's just an artificial way of protecting our calculations from what we don't know (or don't want to mess with) about the extreme UV physics.
Here's an excerpt from pages 866-867 in DeWitt (2003), The Global Approach to Quantum Field Theory (two volumes, Oxford University Press):
In [the Pauli-Villars scheme] one replaces Feynman propagators by differences between Feynman propagators corresponding to fields having different masses. ... It does not correspond to any `vacuum' expectation value and cannot be obtained, with any boundary conditions, from the functional integral for the model. Therefore the Pauli-Villars scheme can in no sense be regarded as a 'physical' modification of a quantum field theory, even a modification that allows states of arbitrarily negative energy.
Regarding the Casimir effect, maybe what Schwartz meant is that the magnitude of the effect is not sensitive to the precise value of the UV cutoff, whether implemented using Pauli-Villars or a lattice. The Casimir effect, at least the simplest version of it, can be modeled using boundary conditions on a free-field model instead of as material plates in an interacting model. In the free-field approach, we can do the calculations exactly with a suitable regulator (like a lattice) and confirm that the force has a finite limit as the UV cutoff is sent to infinity. In this sense, the high-frequency modes don't affect the result. I don't have that book on hand, but I'm guessing that this is probably what Schwartz meant.
The Casimir effect can be calculated using any regulator, and can probably even be calculated without using a regulator in the free-field case. The calculation can also be done using a lattice regulator, and it has a finite limit as the lattice spacing becomes much smaller than the distance between the "plates" (modeled artificially as boundary conditions). Which regulator we use shouldn't matter, because we end up removing the cutoff anyway.
Different regulators may have different advantages in different contexts. Pauli-Villars is nice for preserving gauge invariance, if the symmetries allow mass terms. This is a matter of mathematical convenience, though, not a matter of physical significance.