What is the mathematical difference between the probability density functions of a normal distribution and a standard normal distribution?
The mean and standard deviation
Both are bell-shaped and perfectly symmetrical about the mean.
Besides that, they are identical with respect to the mathematical properties. For example, the Empirical Rule will still hold true regardless if it is a non-Standard Normal distribution. That is, expect about 68% of the distributions to fall within 1 standard deviation of the mean.