What do the dates mean?

On their website FAQs, the Silk manufacturer describe the date on their package:

Our refrigerated beverages and creamers will stay fresh in your refrigerator until the date stamped on the carton. Once opened, they need to be consumed within 7 to 10 days. It is not unusual for our products to remain good past the 10-day mark; however, we only guarantee that they will be good for 7 to 10 days once opened.

If the milk is labelled best by, that is an indicator of quality, not safety. However, normally milk in the US is labelled with a Sell By date, and should be good for a week or so after that date, or for about a week after being opened, whichever comes first.

What are these milks?

Now consider what these two different products are. According to the Go Ask Alice column published by Cornell:

As with all other animal-based foods, it's a complete protein; that is, it supplies people with all the necessary amino acids to form proteins. Cow's milk contains 8 grams of protein and 12 grams of carbohydrates per 8-ounce cup. Cow's milk is a rich source of other nutrients as well. One cup provides adults with about 30 percent of their daily calcium needs and about 50 percent of their vitamin B12 and riboflavin requirements. Often, milk is fortified with vitamin D to facilitate the absorption of calcium. Vitamin A is usually added to milk as well. Depending on the selection, cow's milk can have a significant amount of fat.

[ ... ] One cup of unfortified soymilk contains almost 7 grams of protein, 4 grams of carbohydrate, 4½ grams of fat, and no cholesterol. Although soymilk supplies some B vitamins, it's not a good source of B12, nor does it provide a significant amount of calcium. Since many people substitute soy beverages for cow's milk, manufacturers offer fortified versions. These varieties may include calcium and vitamins E, B12, and D, among other nutrients. If you do choose to use soymilk instead of cow’s milk, read labels carefully to be sure you're getting enough of these important nutrients or consider getting them from alternative food sources.

Milk is a complete food for a growing animal (or human). This makes it an ideal growth medium for microfauna of all types, as it contains everything they need—including easily available sugars to jump start their growth.

Soy milk on the other hand, while derived from the seed intended to grow a full plant, is has a very different profile. It is almost devoid of sugars are readily accessible fats, making it a less ideal growth medium for micro-fauna. This is not say they won't grow, but they may not colonize as readily and grow as quickly.

Conclusion

Therefore, you would expect soy milk to have a longer unopened shelf life than milk as it simply is not as ideal a growth medium. Silk implies it may last longer after opening as well, but being conservative they do not promise that.

Milk will have a shorter shelf life before opening (it is not sterile, and will go bad even in the refrigerator), and definitely will last less time after being exposed to the environment.

They are simply two different products, with very different contents; the fact that both are named "milk" is not an indicator.

See also: Expiration Date on Milk