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
The okara has some unextracted protein and sugar but it also has a large quantity of fiber. That fiber will prevent the protein from properly coagulating. If you leave in all the okara you get soybean porridge.
I imagine you could leave in a portion of the okara and you would get a fragile but hearty tofu. The problem with this is that, with most tofu recipes, the beans are not cooked long enough for the okara to be palatable. With all that sugar it would also give rise to much more flatulence.
You don't have to throw the okara out. There are many recipes that make use of the okara for its fiber. It is pretty straight forward to use it in baked goods. Toasting it gives it more interesting flavor.
If you want to look into tofu-like products that use the entire bean you could research tempeh. The whole beans are fermented so you get a meaty product with all the nutrition of the whole soy bean.
Best Answer
Your title, your problem and your question (btw question is general but complicated) while on the same subject matter are not the same.
In regards to your quest for firm and extra-firm tofu, the OP doesn't state much about the process after the coagulation. Which similarly to cheese making is the (arguably) the most crucial factor in the final texture. firm and extra-firm tofu is pressed with heavier weights, and for longer periods of time.
The choice of coagulent (nikari, gypsum, acid, etc), temperature of the milk and the mixing method can also affect the texture of the final product.