Tears, when not an emotional response, happen when physical stimuli such as irritants offend the senses. Think like a blast of dust or pepper. In these cases the body (correctly) wants the nose to run, in order to irrigate and remove said irritants.
So I guess they're linked. And even though you may be crying over the ending of Ol' Yaller and haven't been tear-gassed by London Riot-Police, or faced-whipped with stink-weed, you still get that river of snot.
I once teared-up and had to blow my nose watching a dust storm from *behind a window*.
"Why does my nose run when I cry? Tears leave the surface of your eye by evaporating or by entering the tear drainage system. A thin tube called the Canaliculus drains the tears from the inner corner of the eyelids down into the nose. Therefore, when your eyes fill with tears, such as when you cry or cut an onion, most of that fluid ends up in your nose."