  The steady state after a quantum quench from the N\'eel state to the
anisotropic Heisenberg model for spin chains is investigated. Two methods that
aim to describe the postquench non-thermal equilibrium, the generalized Gibbs
ensemble and the quench action approach, are discussed and contrasted. Using
the recent implementation of the quench action approach for this N\'eel-to-XXZ
quench, we obtain an exact description of the steady state in terms of Bethe
root densities, for which we give explicit analytical expressions.
  Furthermore, by developing a systematic small-quench expansion around the
antiferromagnetic Ising limit, we analytically investigate the differences
between the predictions of the two methods in terms of densities and postquench
equilibrium expectation values of local physical observables. Finally, we
discuss the details of the quench action solution for the quench to the
isotropic Heisenberg spin chain. For this case we validate the underlying
assumptions of the quench action approach by studying the large-system-size
behavior of the overlaps between Bethe states and the N\'eel state.
