I said this in another comment, but the fundamental difference is that the frequency spectrum in discrete time is periodic and therefore finite in the range that the DAC can recreate, while distortion creates infinite spectra beyond the confines of that period. This is a fundamental limitation of DSP.

The way around it is to make sure the upper harmonics generated by distortion beyond the limit is below the audible threshold, either by making the period larger (over sampling) or modifying the distortion process. It's not lossless and a difficult design problem when factoring in computational limits. It's often cheaper to use discrete analog components with all their flaws than a good-enough processing chain and computer that can evaluate it.

Even offline whitebox modeling tools like SPICE, and realtime black box tools like Volterra series models are limited by this.