Similar, (you are measuring the sugar content of the sample liquid) but the scales are different.
Balling: The name of a density scale for measuring sugar content in water base solutions. Since grape juice is primarily sugar and water, the balling scale was used for a quick and easy "sugar analysis" of juice. The Balling scale contained a slight inaccuracy however, and it was corrected by Dr Brix. Today the Brix scale is in actual use, but the terms Balling and Brix often are used interchangeably.
The Balling (Brix) scale is simplicity itself: Each degree is equivalent to 1 percent of sugar in the juice. For example, grape juice which measures 15.5 degrees on the Balling or Brix scale contains about 15.5% sugar.
Baume: A system for measuring the sugar content of grape juice by its density. It is not easy to use though because the numbers aren't easy to handle: Each degree Baume is equal to approximately 1.75% sugar in the juice. Baume is the more scientific scale, since its based on the specific gravity of a substance.
To confuse it all a bit more deg Plato, deg Balling (Brix) are also the same but the tables differ in their conversion from weight percent to specific gravity in the fifth and sixth decimal place of the specific gravity number.