Here is a blatant cut and paste, it answers your question rather well:
WHY WHITE SUGAR?
Q I am very health-conscious do I have to use white sugar?
A Many health-conscious people are surprised to be told to use refined white sugar in making Kombucha tea and ask whether there is an alternative.
KOMBUCHA NEEDS SUGAR
Just as we need various sugars in order to survive, the Kombucha culture requires sugar and energy, in addition to the minerals and nitrogen it gets from tea, in order for the process of metabolism to take place. The culture cannot provide its own, therefore you have to provide sugar for it. Sugar is used in assimilation and respiration for most of the fermentation, and during its course is broken down and transformed into acids, vitamins, minerals, enzymes and carbon dioxide. Sugar is also involved in the propagation of the Kombucha culture. At the end of the fermentation period, if done correctly, the sugar will have been virtually all converted and will therefore have been rendered harmless.
Various sugars need to be looked at in order to establish which is better to use for the Kombucha fermentation:
Household Sugar (granulated) is refined white sugar and is called sucrose.
Brown Sugar most brown sugars, generally considered a more healthy choice than white, are only refined white sugar which has got its colour from a small amount of caramel or molasses added to it.
Unrefined Brown Sugar this is raw sugar and has a very strong flavour.
Raw Cane Sugar is made mostly into refined white sugar with the remainder steam-heated and sold as pure cane sugar.
Pure Cane Sugar is a healthier alternative to granulated white sugar and contains vitamins, minerals and trace elements etc.
In tests using both unrefined brown sugar and raw cane sugar in the Kombucha fermentation the following results were found:
The solution was dark and cloudy
The taste was quite unpleasant
A poorly-formed culture had formed
There was more yeast sediment
It contained fewer health-giving organic acids
We decided from this, and from other research and information, that unrefined or raw brown sugar was not suitable for the Kombucha fermentation. Refined white sugar either granulated or pure cane sugar is preferable because:
It is transformed during the fermentation process
It provides a good nutrient solution for the metabolism of the Kombucha tea
A healthy culture forms on which to propagate further
It produces a beverage high in organic acids
It makes a good and palatable drink