The biggest advantage is that you get 3X more light sensitivity.

With a bayer filter, every photosite gets 1/3 of the light that falls on it because the filter blocks 2/3 of incoming light to filter for one primary color. So the sensor becomes more sensitive to light. That means that less amplification of the read-out signal to get the same ISO as with a conventional sensor. The end-result is that you get lower noise at each ISO sensitivity.

There is no need for an Anti-Aliasing filter, so you get better sharpness and micro-contrast. As Nikon proved it though, this is not necessary for Bayer-based cameras but is usually the case. When a camera uses an AA filter, it blurs the light before it reaches the sensor to avoid the occurrence of an artifact called moire. Any time you blur something, you reduce contrast because you spread light over multiple pixels. Without an AA filter, the blurring does not happen and you get better contrast.

B&W sensors also obviously do not need Bayer-interpolation. This means that the readout is the image data and there is no question of softness introduced by interpolation (or the AA filter that is not there) and no need to sharpen at the capture level, although you may sharpen when processing for your output medium (print, screen or other).