It has been noted that elephants trained for war, as was done occasionally in earlier times, have still shown a tendency to panic in battle much more often when compared to a war horse trained for the same purpose. However, is this due to an innate quality of elephant behavior, or a result of poorer training methods due to humans simply having less experience training elephants than horses? Are elephants naturally more fearful creatures than horses or other beasts of war?
I doubt if a comparison between these two animals in war situation is in order. These two animals serve different roles.
Here is a reference to an article in Wikipedia on War Elephants. Elephants have been used in war for thousands of years pretty effectively till the enemies discover their weakness and use it to their advantage. Making the elephant panic is one of the ways to counter them. Here is a list of methods from the above article:
Elephants had a tendency to panic themselves: after sustaining painful wounds or when their driver was killed they would run amok
One famous historical method for disrupting elephant units was the war pig......
At the Megara siege during the Diadochi wars, for example, the Megarians reportedly poured oil on a herd of pigs, set them alight, and drove them towards the enemy's massed war elephants. The elephants bolted in terror from the flaming squealing pigs.
It is for sure that many horses also panic in war situations. However a panicked elephant can cause more havoc than a horse. It appears that to prevent a panicked elephant from running amuck back into the ranks
The driver, called a mahout, was responsible for controlling the animal. In many armies, the mahout also carried a chisel-blade and a hammer to cut through the spinal cord and kill the animal if the elephant went berserk.
In recent times elephants in temple festivals in india do occasionaly panic and run amuck when the high explosive fire works are set of. However most of the elephants seem to tolerate it.