There is no difference between the 2 biologically - they are one and the same. Brown rice has the bran left on, white rice has had it removed through polishing (the husk is removed from both types of rice). Just like white or wholemeal flour...
Brown rice needs longer to cook before it takes more time for water to get into the rice with the bran on it, and it is also more fibrous and is digested slower so gives less of a carbohydrate (sugar) rush. And as a result of the husk and being more fibrous is considered better for you because of the higher fibre content. I'm not certain if there are more calories or nutrients in brown rice as well, but would guess that at the minimum there are more nutrients.
Brown rice
apparently contains more magnesium, manganese, calcium and zinc (etc) than white rice, so I guess that is also found in the 'brown shell' that is under the husk, but this is where life gets interesting.
Taking the United States Department of Agriculture Agricultural Research Service National Nutrient Database figures from the following sites...
Brown rice, medium grain cooked,
http://ndb.nal.usda.gov/ndb/foods/show/6456
White rice, medium grain cooked, unenriched.
http://ndb.nal.usda.gov/ndb/foods/show/6573
... I get this which I did as a pdf which compares them.
But I should point out that even though brown rice has roughly 2-5 times the quantity of vitamins and minerals, the quantities that are in either brown or white cooked (unenriched, medium grain for the sake of comparison) rice are so low per 100g of cooked rice (roughly 1/2 cup) that neither comes close to really even helping to provide you with the RDA (recommended daily allowance) of these vitamins and minerals... for example - 100g of cooked white rice (unenriched) has 3mg of calcium in it; the same brown rice has 10mg - so 3 times as much.. great but the
RDA for calcium 1,000mg.