They both come from the same plant, but, as with red and green peppers, the final color has to do with their ripeness. Peppercorns are the berries of the pepper plant (piper nigrum), which is native to Southern Asia. You might be interested to know that the primary spice that Europe's explorers were seeking when they accidentally found the New World on their "shortcut" to Asia was pepper. It still accounts for a quarter of the world's spice trade.
A black peppercorn is picked when still green and dried in the sun until it turns black. A white peppercorn ripens fully on the vine before it is picked. Black pepper has a slightly hotter flavor and aroma. As with any spice, if you grind the pepper as you use it, it will have lots more flavor than if it was ground in a factory months ago and sat on the shelves in the grocery store before it sat on your shelf. So the question of how much to use is very much a matter of personal preference.
It is unclear to us why a recipe would call for both white and black pepper - apparently the writer found the ideal combination of ingredients to suit his tastes. Often white pepper is called for in a recipe for a light-colored food or sauce for aesthetic reasons.