The camera converts to B&W in one way only -- typically taking the relative brightness of each pixel and using that as an indicator of "how much black vs. white".
Converting using the channel mixer lets you decide how much of each red, green, and blue channel is used in the conversion. This lets you achieve MUCH more dramatic effects -- for example, you can use 0% of the blue channel, and a blue sky will appear basically black (with very white clouds still). This can be applied for almost any color you like, and in more subtle ways. As just on example, careful mixing can increase the apparent contrast of the image, without blowing out entire light or dark regions.
In short -- automated conversion will give you an image with no thought for artistic quality -- it's fully automated. Using the channel mixer allows you to make the artistic decisions -- after all, you probably wanted to convert to B&W because of its artistic merits in the first place!