Brown rice has more healthful attributes to offer our bodies than its white counterpart. To start, it is less refined. Brown rice is the whole rice grain that only has its husk/hull (the outermost layer of the grain) removed during the milling process. It still contains its bran (fiber) and germ, both of which provide many nutrients and health benefits.
White rice is created when brown rice is further milled and the bran and germ is removed, thus reducing its fiber and nutrient content. The "polishing" process removes what's called the "aleurone layer" of the rice, which contains essential oils that support good health - but you'll only get those oils from brown rice, and not white. White rice does have a couple of redeeming qualities that can make it an important choice for certain people and populations: it has a longer shelf life and is easier to digest than brown rice. When either of these factors is of essential importance, either due to food storage conditions or a GI issue, white rice may be a better choice than brown.
Brown rice is less processed than white - and as the general rule of thumb goes, the less processed a food (and closer it is to its natural state), the better it is for you. Brown rice has more fiber, more protein and a lower impact on your glycemic index than white rice - all of which contribute to it likely making you feel fuller longer than an equal amount of white rice - and make it the more healthful choice. On a more superficial note, it also has a richer and slightly nuttier taste and texture, making it a wonderful and dynamic ingredient to cook with in a number of recipes, both savory and sweet (try making brown rice risotto, a stir-fry with brown rice, or brown rice pudding!).
You can learn more about the benefits of brown rice via the following resources: