The difference between the $\mu$-problem and the hierarchy problem is that loop corrections to the value of $\mu$ in MSSM are small and convergent, because of supersymmetry, while the loop corrections to $m_h^2$ in the SM are divergent. So to explain why $\mu$ is small, it is enough to explain why its approximate – tree-level – value is small. (Well, the superpotential doesn't receive any perturbative corrections whatsoever, but I don't want to get to the interesting realm of stronger statements here.) Once you explain or assume the value of the tree value, the theory is OK.
On the other hand, to explain why $m_h^2$ is small, one always needs to work with the full value including all the loop corrections which are large. So in the SM case, it's much easier to convince oneself that the almost exact cancellation of the Higgs mass is a coincidence that doesn't happen naturally. In the MSSM case, a low value of $\mu$ is a feature that may occur naturally.
The coefficient $\mu$ in $\mu H_u H_d$ term in the superpotential shouldn't be much heavier than the electroweak scale because it determines the masses of higgsinos; and it contributes to the Higgs bosons' self-interactions. Higgsinos that are vastly heavier than the actual Higgs mass would mean that the SUSY breaking (via higgs-higgsino mass difference) is extremely big and a big part of the hierarchy problem of the SM would re-emerge in the MSSM. Too strong interactions of the Higgs bosons could also be a problem.
