Since the part about the heart rate increasing and blood flow carrying venom around the body at a faster rate has been covered, I would suggest one more way that the victim who is terrified and hopeless will die more quickly than a calm one.
As the previous posts mentioned, the sympathetic nervous system produces a huge response with adrenaline and the heart rate increases, all those things happen and they actually make it significantly more difficult to think carefully and calmly about what you should do about this particular snake bite. This panic itself can make it harder to survive because it may be more difficult to recall any training or previous experience that would help the victim in this case.
Suddenly becoming a snake bite victim can certainly be an anxiety riddled situation. When we are anxious or scared the heart rate increases, the increase in rate causes the heart to pump more blood to distant parts of the body faster than normal. These events take place due to the sympathetic nervous system.
If a snake bite victim is scared or terrified the heart will beat faster, if the snake was venomous and if the venom was injected into the circulation, the fast heart rate will carry the venom to the rest of the body faster. The best chances of survival to a victim of a venomous snake bite are realized when the victim can try to remain calm and still, thus not elevating the heart rate. Any movements of the body will also act to increase the heart rate and propel the venom through the circulation faster.
The most glaring and obvious effect of being scared or terrified when one has been bitten by a poisonous snake is that, when you are scared, your heart rate increases and your blood pumps faster. This is what is supposed to happen because we get scared when there is danger to our safety, and we need extra oxygen to fight or flee. This is how humans have survived since the beginning.
In this case, though, that works against you, as the faster your blood pumps, the faster the poisonous venom from the snake reaches the heart and the brain, whereas a person who is calm has a slower pulse, and more time before the bite becomes fatal, and therefore more chance to get help and survive.