Often the dairies producing organic milk are closer to the point of delivery, so it doesn't travel as far or sit in warehouses as long as regular milk. That makes it fresher, for one thing, but also extends the length of time before purchase that it will still be good.
Once a carton of milk is opened, however, the shelf life begins to decrease for all kinds of milk.
Some people believe that the addition of hormones and antibiotics in cows that produce regular milk cause it to "go bad" faster. It's easy to see why organic milk tastes better than regular (because it doesn't contain anything that doesn't taste natural) ~ but the subtle effects of not adding chemicals goes far beyond the taste. It's actually better for your health.