One cup of white sugar equals 200 grams, while 1 cup of packed brown sugar equals 220 grams. The measurements are different because ounces, teaspoons, tablespoons and cups are all measurements of the volume of a substance, while grams measure a substance's weight.
The weight of solids often differs, even if their volumes are the same. For instance, a cup of sugar and a cup of flour are both the same volume, but they aren't the same weight; a cup of flour weighs 75 grams less than a cup of sugar. This is also true for liquids. It takes about 240 grams of a liquid such as a beverage to equal 1 cup, but this is still not an accurate measurement for heavier liquids or for soups that also contain solids.