gaussian window for designing filters in DSP
First, a Gaussian window will have some parameter controlling the ‘width’ of the Gaussian pulse.  So there is no singular Gaussian window like there would be with many other window types.
Second, because it is a window, it has a defined beginning and end such that the signal function is zero outside the window. The is mathematically equivalent to multiplying your Gaussian function with a rectangular window  this means the resulting frequency response is the convolution of the gaussian frequency response and the rectangular frequency response, which will contain side lobes (ringing). The magnitudes of the side lobes are a function of the window length and the pulse width.  Whether or not the side lobes are better or worse than, say, a Hanning window depends on how you define those parameters and what your metrics are.
Lastly, and this wasn’t explicitly called out but is probably relevant, is discritization. If you are using the resulting window function as part of a discrete time process (in software, for example) the number of samples in the window will effect the frequency response. Every continuous time window function has an infinite bandwidth frequency response because the time domain response is limited. As such, aliasing will occur and the discrete time frequency response will change. The amount it changes is minimized by increasing the number of samples in the window, either by increasing the window length, or increasing the sample rate.