Drawing with white charcoal
No, you don't really need that white charcoal :) However, it can be a very useful and fun to work with.
White charcoal can be used to create highlights in drawings, and is particularly effective in combination with dark charcoal. It is often used in academic drawing when drawing from a real-life references, like nudes:

Pierre-Paul Prud'hon - Standing female nude, seen from behind - 1810s
Notice the hue of the paper: white charcoal is naturally especially useful when used on non-white paper, preferably dark paper, as mentioned in the comments by users Lyssagal and agarza, or, when used in combination with charcoal, a hue in between white and black (medium greys, often slightly coloured, in blue, green, or yellow, are quite common).
Using this approach, you can use white charcoal as an opposite of charcoal, in the sense that you build your subject up from the illuminated aspects instead of its shadows, as one usually does using charcoal. Together, especially on toned paper, these two complimentary approaches give you more freedom, a larger tonal depth, and a quicker method of working, as you can correct most (minor) errors on-the-fly.
You can actually create your own toned paper by crushing some charcoal and rubbing it on a paper sheet using the palm of your hand (if necessary, you can use tissue or cloth to even it out more).
Unlike charcoal, white charcoal is harder to erase, and less prone to blurring as regular charcoal is (see the note below for the reason).

Note that - unless you're talking about binchō-tan - 'white charcoal' is obviously a misnomer: charcoal, from 'charred coal', is the product of burning wood.
This so-called &quot;white charcoal&quot; may be similar in character and regularly used in conjunction with charcoal, but is a pigment combined with a binding agent, often a (compressed) soft pastel. Binchō-tan is indeed (seemingly) white charcoal, but not suitable for drawing (and would probably appear grey).