For raw milk drinkers the benefits of having a delicious, easy to consume food that is rich in amino acids, immunoglobulins, probiotics, as well as fat, protien and energy, may seem like enough.
But if you are already budgeting for a high quality product, it's totally worth it to maximize the nutrients you are getting in your food. And that's where pasture comes in.
There are three main benefits to drinking milk from cows eating pasture vs. conventional high-grain diets.
Benefit 1: Increased vitamin content in the milk.
In general, the more grain a dairy cow is fed, the more milk she will produce--but the vitamin content doesn't increase in the same way. That means when you get milk from a cow that gets more grain and makes tons of milk, you are getting less of the vitamins and more water in your milk. Milk from cows fed mostly pasture may be more expensive for this reason (the farmer has a higher cost per gallon), but the consumer is getting a lot more vitamins per glass. In particular, you will get more vitamins A, C, E, and both forms of K (K1 and K2).  
Benefit 2: WAY more beneficial fats.
Grassfed cows produce milk with a lot more CLA in it than cows on pasture part time or not at all. According to one study, "Cows grazing pasture and receiving no supplemental feed had 500% more conjugated linoleic acid in milk fat than cows fed typical dairy diets [high in grains]."  CLA is thought to reduce body fat, decrease cardiovascular diseases and cancer, improve bone mass, increase immune system health, and decrease autoimmune reactions. Pretty cool!
Benefit 3: Planned grazing = Healthy soil and healthy plants = more nutrients!
Planned grazing with ruminants (cows, sheep, goats, etc.) can have amazing beneficial effects on soil, plant, and animal biodiversity . Over time, this increase in soil and plant health means that animals eating the plants are healthier, and produce more nutrient-rich foods. The soil is healthy and crops grown in it will also have more nutrients. All without relying on chemical fertilizers. To me, that's true food security!
 Jensen, S. K., A. K. Johannsen, et al. (1999). "Quantitative secretion and maximal secretion capacity of retinol, beta-carotene and alpha-tocopherol into cows' milk." J Dairy Res 66(4): 511-22
 Fu, X., Harshman, S. G., Shen, X., Haytowitz, D. B., Karl, J. P., Wolfe, B. E., & Booth S. L. (June 2017). Multiple vitamin K forms exist in dairy foods. Current Developments in Nutrition. doi.org/10.3945/cdn.117.000638
 Dhiman, T. R., G. R. Anand, et al. (1999). "Conjugated linoleic acid content of milk from cows fed different diets." J Dairy Sci 82(10): 2146-56