How does diabetes cause neuropathy and extreme thirst?

I have read that diabetes is associated with neuropathy (peripheral neuropathy and autonomous nervous system disorders such as urinary tract malfunction), as well as thirst. Specifically, how does it cause these two symptoms? It's OK if the answer is complicated; I'm just looking for an accurate explanation, since most of the web pages I've found just say that these are symptoms of diabetes, without explaining why.

Those are really phenomenal answers! Thanks so much. I should add that the reason I asked is that I have both symptoms, but my blood sugar is normal and any other diabetes tests were negative. I also have a few other minor symptoms of diabetes. Could these symptoms still have a common cause? (My doctor can't figure it out.)

Well extremely high blood sugars causes a degradation of the myelin sheath of the nerve cell. This cause the nerve impulse to move more slowly and in some cases not at all.
As for why it make you thirsty the body always wants a balance of water inside and outside of the cells. When you have high blood sugar it causes water form inside the cell to move outside. Causing you to become thirsty

(Assuming you are referring to type I, or insulin-dependent diabetes):

Sugar is required by cells to have energy for all life functions. Sugar cannot cross the outer cell membrane on it's own because of it's chemical properties and size. Therefore, in order to enter the cell to be used for energy, it needs another chemical to attach first and "open" the door to the cell for it. This other chemical is called insulin, and is secreted by the pancreas.

In a diabetic, the pancreas does not secrete insulin anymore due to damage to the beta-cells of the pancreas. Without insulin, no sugar is absorbed into the body's cells, and you get a massive overload of sugar in the bloodstream. This kicks the kidneys into high gear to reduce the sugar in the blood by urinating it out, which also takes water with it.

The end result is that the body becomes dehydrated severely by excess urination. As with any normal dehydration condition, the thirst center in the hypothalamus of the brain is activated, and you become very thirsty. This happens as the hypothalamus uses its osmotic receptors to detect the concentration of the blood.

As for why it leads to neuropathy, the brain consumes monstrous amounts of sugar to function. When the body is insulin deficient, the brain cannot absorb enough sugar to keep it alive, and parts of the brain can actually die off or malfunction. The brain is the body's largest consumer of sugar, so it is the first organ to likely go south when sugar cannot be absorbed.