The USA Rice Federation provided a comparison table for many types of rice:

The information in this table was taken from the U.S. Department of Agriculture, Agricultural Research Service, 2002. USDA Nutrient Database for Standard Reference, Release 15 (August, 2002).
For the measure of 46.25 grams grams of brown and white long-grain cooked rice, you can conclude from the table that:
- Brown rice is 2.7x richer in dietary fiber than white rice.
- White rice has almost same number of carbohydrates with brown rice (negligible).
- The number of calories is almost the same
- For minerals, the brown rice is richer than white rice.
Now, as you see, brown rice and white rice have similar amounts of calories and carbohydrates, so there might not be a huge difference after all, right? Not really, the main differences between the two forms of rice lie in processing and nutritional content, researched by a Wikipedia page:
When only the outermost layer of a grain of rice (the husk) is removed, brown rice is produced. To produce white rice, the next layers underneath the husk (the bran layer and the germ) are removed, leaving mostly the starchy endosperm.
Several vitamins and dietary minerals are lost in this removal and the subsequent polishing process. A part of these missing nutrients, such as vitamin B1, vitamin B3, and iron.
Sometimes, they are added back into the white rice making it "enriched", as food suppliers in the US are required to do so by the Food and Drug Administration.
One mineral not added back into white rice is magnesium; one cup (195 g) of cooked long grain brown rice contains 84 mg of magnesium while one cup of white rice contains 19 mg.
When the bran layer is removed to make white rice, the oil in the bran is also removed. Rice bran oil may help lower LDL cholesterol.
Additionally, researchers at the Harvard School of Public Health reported that eating two or more servings of brown rice weekly seems to be associated with a lower risk of developing type 2 diabetes. They also report that eating five or more servings of white rice per week is associated with an increased risk.
The scientists examined rice consumption and diabetes risk in 39,765 men and 157,463 women in three large studies. They analyzed responses to questionnaires completed every four years about diet, lifestyle, and health conditions.