Bell (1964) writes that if we assume an equivalent classical hidden variable distribution for a two-qubit state then the expectation value of the product of two observables $A$ and $B$ can be written as $$P(\vec a,\vec{b})=\int d\lambda\, \rho(\lambda) \, A(\vec{a}, \lambda)\, B(\vec{b}, \lambda)$$ where $\rho$ is the probability associated with the hidden variable $\lambda$ and measurements are made along the vectors $\vec a$ and $\vec b$ on $A$ and $B$ respectively.
When we formulate the Peres-Horodecki separability criterion for density matrices, we write the separability of the density matrix as $$\rho^{AB}=\sum_i p_i \rho_i^A \otimes \rho_i^B$$ which gives the following result for the quantum operators (forgive the terrible notation): $$<A \otimes B>_{\rho^{AB}}\,=Tr [(A \otimes B) \,\rho^{AB}]=\sum_i p_i Tr[A\rho_i^A]\, Tr[B\rho_i^B]$$ which appears to be just a discrete analogue of the Bell integral ($d\lambda\, \rho(\lambda)$ corresponds to the weight $p_i$).
So what is the physical difference between the two statements which leads to a difference in their predictions?