The only times I've had deer or elk drop instantly the bullet or arrow caused either complete nervous system disruption (brain shot or severed spinal cord in the neck), or caused a calm animal's blood pressure to hit zero instantly (severed aorta). My brother, who is a doc, tells me that the latter causes instant death in humans, too, and is a common way that surgical mistakes kill patients so quickly that they can't be revived. Surprisingly, heart-shot animals often run quite a ways, even when the heart is completely destroyed.
With all other wounds - no matter how severe or what weapon, arrow, or bullet was used - animals can be amazingly tough. I shot my biggest bull elk from 12 yards away with a huge 3-blade broadhead. He was quartering towards me, and that broadhead took out the near lung, the diaphram, the liver, passed all the way through the elk's abdomen, went through all the hip muscles on the far side, and wound up just under the skin. That elk still ran 800 yards, and I had one heck of a time finding him - I was at 11,000 feet in the mountains north of Taos, NM, and it started raining and snowing right after the shot, which covered tracks and blood trail. Found him the next morning by traversing back and forth across the mountain side, dropping down 20 vertical feet after each traverse. I've had deer that were hit perfectly with a high powered rifle run well over 100 yards. On the other hand, I've deer and elk that were perfectly hit through both lungs drop after running only 20-30 yards.