Baking soda needs acidic ingredients, such as cream of tartar, to release carbon dioxide and cause batter to leaven in a baking recipe. Baking powder, on the other hand, which already contains the acidic ingredients to induce a carbon dioxide producing reaction, doesn't need any other low-pH ingredients to activate.
Baking powder boosts the rising power of baking soda, which is why certain recipes call for both. Additionally, they have different effects on browning and flavor; baking soda helps foods brown better, and baking powder can help maintain tartness in a recipe.
Source: Flickr User Mel B.