The real importance- and "uniquness"- of the Normal distribution is this: if we have a large number, n, of trials fromanyprobability distribution (with finite moments) the probability distribution for the mean of the trials is approximately Normal with the same mean and with standard deviation equal to the standard deviation of the original distribution over the square root of n. That is, if you take a large enough sample the Normal distribution will work no matter what the "underlying" distribution is.