  The mass of the lightest CP-even Higgs boson of the minimal supersymmetric
extension of the Standard Model (MSSM) has previously been computed including
O(alpha alpha_s) two-loop contributions by an on-shell diagrammatic method,
while approximate analytic results have also been obtained via
renormalization-group-improved effective potential and effective field theory
techniques. Initial comparisons of the corresponding two-loop results revealed
an apparent discrepancy between terms that depend logarithmically on the
supersymmetry-breaking scale, and different dependences of the non-logarithmic
terms on the squark mixing parameter, X_t. In this paper, we determine the
origin of these differences as a consequence of different renormalization
schemes in which both calculations are performed. By re-expressing the on-shell
result in terms of MS-bar parameters, the logarithmic two-loop contributions
obtained by the different approaches are shown to coincide. The remaining
difference, arising from genuine non-logarithmic two-loop contributions, is
identified, and its effect on the maximal value of the lightest CP-even Higgs
boson mass is discussed. Finally, we show that in a simple analytic
approximation to the Higgs mass, the leading two-loop radiative corrections can
be absorbed to a large extent into an effective one-loop expression by
evaluating the running top quark mass at appropriately chosen energy scales.
