Some applications such as identification or Monte Carlo based uncertainty quantification often require simple analytical formulas that are fast to evaluate. Approximate closed-form solutions for the natural frequencies of free orthotropic plates have been developed and have a wide range of applicability, but they are not very accurate. For moderate ranges of plate parameters, such as those needed for identifying material properties from vibration tests, good accuracy can be achieved by using response surface methodology combined with dimensional analysis. This paper first demonstrates that such a response surface can be much more accurate then the approximate analytical solutions even for relatively large ranges of the material and geometric parameters. Second it compares the accuracy of the elastic constants identified from experiments using the two approximations. It demonstrates the advantage of high fidelity approximations in vibration-based elastic constants identification. For a least squares identification approach, the approximate analytical solution led to physically implausible properties, while the high-fidelity response surface approximation obtained reasonable estimates.