The basic formulation of the least-squares method, based on the L2 norm of the residuals, is still widely used today for identifying elastic constants of aerospace materials from experimental data. While this method often works well, methods that can benefit from statistical information, such as the Bayesian method, may sometimes be more accurate. We seek situations with significant difference between the material properties identified by the two methods. For a simple three-bar truss example we illustrate three situations in which the Bayesian approach systematically leads to more accurate results: different sensitivity of the measured response to the parameters to be identified, different uncertainty in the measurements, and correlation among response components. When all three effects add up, the Bayesian approach can be much more accurate. Furthermore, the Bayesian approach has the additional advantage of providing the uncertainty in the identified parameters.We also compare the two methods for a more realistic problem of identification of elastic constants from natural frequencies of a composite plate.