Here is a histogram of the daily closing differences for General Electric. On top of it is superimposed a fit of the histogram with first the Normal distribution and then the Cauchy distribution. The fit was found using maximum likelihood estimation.
One of these is a better fit. Which do you think?
Of course this is not proof that the differential is Cauchy distributed. For that however you have to simply look at the properties of each distribution. For example the sum of two Cauchy random variables is another Cauchy random variable with parameters equal to the sum of the two previous parameters. Define beforehand the properties of stocks and you can derive the behavior of the distribution which should match it.
Using the normal distribution is fine if you are making some kind of approximation. However whenever an approximation is made, you have to ask how good it will be and under what conditions it fails. It looks like this hasn´t been seriously tried until recently.