In this context, the black hole entropy is the fine-grained entropy of the black hole. Namely, if you knew all black hole microstates and hence the density matrix of the black hole, $\rho_\text{BH}$, the black hole entropy would be the von Neumann entropy associated to $\rho_\text{BH}$
$$S_\text{black hole} := S_\text{vN}(\rho_{\text{BH}}):= -\mathrm{tr}(\rho_\text{BH}\log\rho_\text{BH}).$$
The Bekenstein-Hawking entropy is the coarse-grained, or equivalently, the thermodynamic entropy of the black hole.
$$ T\,\mathrm{d}S_\text{Bekenstein-Hawking} \overset{!}{=} \mathrm{d}M - \Omega\,\mathrm{d}J.$$
Knowing $\rho_\text{BH}$, this is equivalent to the following. Choose a bunch of coarse-grained observables that you want to measure $\left\{\mathcal{O}_i\right\}_i\subset\left\{\text{all observables}\right\}$ and find all density matrices $\rho_\text{BH}^\text{coarse}$ such that
$$ \mathrm{tr}(\rho_\text{BH}^\text{coarse} \mathcal{O}_i)=:\left<\mathcal{O}_i\right>_\text{coarse}\overset{!}{=}\left<\mathcal{O}_i\right> := \mathrm{tr}(\rho_\text{BH}\mathcal{O}_i), \qquad \forall\mathcal{O}_i\in \left\{\mathcal{O}_i\right\}_i.$$
Then the Bekenstein-Hawking entropy is
$$S_\text{Bekenstein-Hawking} := \max_{\rho_\text{BH}^\text{coarse}} S_\text{vN}(\rho_\text{BH}^\text{coarse}).$$
This turns out to be exactly what we, unknowingly, do when we compute thermodynamic entropies.
This definition also makes it clear why
$$ S_\text{black hole}\leq S_\text{Bekenstein-Hawking}.$$