What is the advantage of Wiener filter for noise reduction of a 1D signal?
The Wiener filter considers statistical behaviours of the noise and the signal, and thus, (theoretically) achieves optimum separation of them for a class of signals and systems, which is not the case for more classical approaches.
Wiener filter frequency response is such that, at those frequencies where noise power is dominant (a.k.a. low SNR), the gain is reduced, and the output is suppressed; causing noise-reduction. Whereas for frequencies at which the signal power is dominant (a.k.a. high SNR), then the gain is closer to one, and output is closer to the input.
Thus, to determine the gain of the Wiener filter, at a particular frequency, you must know the Power Spectral Densities (PSDs) of the noise and the desired signal.
Typically, you won't know these in advance very well; but they can be estimated from available data. The result is a departure from the optimum (ideal) performance.