The White Balsamic is made with Sweet White Trebbiano grape yes, but so is the dark balsamic.
The Dark Balsamic gets the dark color from the sugars caramelizing during the cooking process while it is being reduced and concentrated.
The White Balsamic is reduced in a vacuum chamber. This allows the water content to evaporate at a much lower temperature, below the temperature that sugar of the grape must will caramelize at. White wine vinegar is used instead of red wine vinegar and it is not put in wood barrels. The taste is a little lighter and less bitter without the caramelization. Even though the sugar content is the same as the dark, it can actually taste sweeter because the bitter does not offset the sweetness.
The flavors of the two are very similar, although the dark balsamic is slightly sweeter and tends to be a little more syrupy. The white has more of a clean aftertaste. The main reason one would use white balsamic, rather than regular, is mostly aesthetic. It can be used with lighter colored foods, dressings, or sauces without any discoloring. If that sort of thing matters to you.