The approximations don't just introduce small errors. To simulate quantum systems classically, you need to make drastic assumptions that fundamentally change the nature of the system.
This is very different from, say, an approximation that adds in a small amount of noise that you can estimate. The approximations in simulating quantum systems classically can radically change the behavior of the system, in ways that you might not understand or be able to easily estimate.
However, quantum computers also have to deal with noise and errors. So far, that's not very different.
(If we manage to build error-correcting quantum computers, that might change.)