Some of us may have some idea how sugar and molasses are interrelated. Basically, molasses is the by-product of the sugar refining industry. In most parts of the world cane is the primary source of both the sugar and the resultant molasses, but in some areas sugar beet is used for the process instead. A batch of cane juice will be boiled in stages to promote the crystallization of the sugar; typically until the third boil consequently yields the thickest, darkest syrup with the maximum concentration of minerals and lowest sugar content.
Brown sugar also comes in various forms; it is essentially sugar with differing amounts of molasses. Natural brown sugar, sometimes referred to as 'raw sugar' is what results from the first crystallization of cane juice. It has a high sugar content, but also contains some minerals due to the molasses present within the crystals. Even natural brown sugar comes in different grades, of which 'Muscovado' is the absolute least refined. Turbinado is a grade of natural brown sugar that has been spun in a centrifuge to remove water and impurities. Most other forms of 'brown sugar' are made by mixing fully refined sugar with molasses.
In conclusion, blackstrap molasses is the best bet for a carbohydrate supplement with a good mineral content; even the most natural brown sugar is only partially molasses and has much less minerals.
You can compare for yourself here, just make sure you set each serving size to the same sample (ie: 100 grams) and also remember that actual mineral contents vary.
Molasses:
http://nutritiondata.self.com/facts/sweets/5573/2
Brown sugar:
http://nutritiondata.self.com/facts/sweets/5591/2
If you're ever particularly curious about something (such as the nutrient content of x), remember that you're on the internet!