zoology - Is there a difference between pig and cow anatomy that makes religious slaughter of the latter less painful?

This question is not for the squeamish. I was told that non-Kosher animals like pigs have a different anatomy in their neck compared to kosher animals like cows such that the Shechita method of religious slaughter (a very sharp knife held on the neck only by the weight of gravity rapidly going back and forth in a single cut that drains blood and results in an "instant" drop in blood pressure in the brain), while causing pain in the non-Kosher animals, doesn't cause pain in the Kosher animals. Is there such an anatomical difference and thus difference in pain?

(If this would be more on-topic at Skeptics.SE, let me know.)