  In a previous paper (hep-th/0509071), it was shown that quantum 1/J
corrections to the BMN spectrum in an effective Landau-Lifshitz (LL) model
match with the results from the one-loop gauge theory, provided one chooses an
appropriate regularization. In this paper we continue this study for the
conjectured Bethe ansatz for the long range spin chain representing
perturbative planar N=4 Super Yang-Mills in the SU(2) sector, and the ``quantum
string" Bethe ansatz for its string dual. The comparison is carried out for
corrections to BMN energies up to 3rd order in the effective expansion
parameter $\tl=\lambda/J^2$. After determining the ``gauge-theory'' LL action
to order $\tl^3$, which is accomplished indirectly by fixing the coefficients
in the LL action so that the energies of circular strings match with the
energies found using the Bethe ansatz, we find perfect agreement. We interpret
this as further support for an underlying integrability of the system. We then
consider the ``string-theory'' LL action which is a limit of the classical
string action representing fast string motion on an S^3 subspace of S^5 and
compare the resulting $\tl^3/J^2$ corrections to the prediction of the
``string'' Bethe ansatz. As in the gauge case, we find precise matching. This
indicates that the LL Hamiltonian supplemented with a normal ordering
prescription and zeta-function regularization reproduces the full superstring
result for the $1/J^2$ corrections, and also signifies that the string Bethe
ansatz does describe the quantum BMN string spectrum to order $1/J^2$. We also
comment on using the quantum LL approach to determine the non-analytic
contributions in $\lambda$ that are behind the strong to weak coupling
interpolation between the string and gauge results.
