Thresholding a difference image with a very high kurtosis
The first thing that jumps to my mind is that you want to test if your data is significantly different from zero. Determining if it is positive or negative should be trivial.

I would set a threshold based on the standard deviation of the results. Anything greater than say $\pm2\sigma$ from the mean (zero) are considered to be significant changes. The suitable value of sigma to choose on how peaked your distribution is and how many and how big changes you expect.

This sort of approach is traditionally used to look for outliers, conventionally with $\pm3\sigma$, which is quite similar to what you want.