Accurate measurement of object's position on a bayer-filtered and grayscaled images
The key here is how well your acquisition system can differentiate the ball from everything else.  The other key is that the larger number of pixels on the object the better.

From a look at the video, and without seeing the grayscale version, I would think that a color camera (with the same pixel density as the greyscale camera) would do better.  

One trick is:  greyscale cameras are often more sensitive to IR and, as a result, have an IR filter inserted just above the image sensor.  Your ball might be close enough to that so that, by removing the IR filter, the greyscale camera would do as well at seeing the ball as the color camera.

Another trick is:  Try illuminating the field of view of either camera with a different colored light.  You may find that the ball shows up better with one light source over the others (in one camera or the other).

Finally: try zooming in on the ball.  That will get you more pixels and either the color or greyscale camera will benefit from the extra pixels.