A remarkably terse specification of the instrumentation (sensors, probes) and techniques used to measure "the vibration of a structure" leaves open the issues of applicability of the Wiener filter vs "the other (more classic) filtering methods" to processing of "the-vibration-of-a-structure" data.
Well established techniques are not readily amenable to classification on a list of unconditional advantages/disadvantages. Rather, one can compose the lists of features, itemizing the relevance of the solution for a set of application scenarios. The Wiener filter is adaptive, and this feature makes it well suited in changing environments. On the other hand, being an estimator, the Wiener filter guesses at the denoised signal waveform, but not without restrictions: it minimizes MSE when both processes, a signal of interest and the noise, are Gaussian; it implements the linear signal processing model. To implement a casual filter variety -- the classic Wiener filter -- the additional measurements are needed to trace the signal/noise statistics.
Depending on application, one may have to consider the other techniques, like nonlinear estimators, ARMA, recursive estimation solutions, or even the Kalman filter (predictor/corrector), although the latter is not adaptive in its basic implementation. But it is not a recommendation: the OP provides too little information to advise them on the preferential SP techniques and write down "the equations".