Simulating a noiseless quantum circuit classically is as simple as calculating:
$$\tag{1}
|\psi(t)\rangle = e^{-\frac{\textrm{i}}{\hbar }Ht} |\psi(0)\rangle,
$$
where $|\psi(t)\rangle$ and $|\psi(0)\rangle$ are vectors each with $2^n$ elements (for an $n$-qubit circtuit) and $H$ is a Hermitian matrix with $2^n \times 2^n$ elements.
When you add noise, the qubits of the circuit can (and usually do) get entangled with their environment, meaning that a wavefunction is no longer good enoough to describe the statistics of the system of qubits, and a density matrix is needed instead. Even for very weak noise where a Markovian master equation is good enough, that Markovian master equation will be significantly more complicated than Eq. 1, and will involve the density matrix $\rho(t)$ which has $2^n \times 2^n$ elements rather than only the $2^n$ for a wavefunction. If you want to see what the dynamics of $\rho(t)$ looks like for a Markovian master equation, you can see my answer here.
As the coupling gets stronger, a weak-coupling Master equation or Redfield equation is not enough, and you may need to do something more accurate such as numerically calculating the double Feynman integral I described in my answer to: Simulating a quantum circuit with decoherence and noise.
The answer by Ron Cohen explains that as noise increases, more qubits may be needed for quantum error correction (QEC), but you're correct that QEC is usually not implemented in NISQ devices, and therefore the point is that a noise-resilient quantum circuit will be easier simulated by the unitary evolution of Eq. 1 or by a weak-coupling Markovian master equation rather than a numerical Feynman integral or other non-Markovian approach.
