It is important to understand the difference between these two sentences if you want to master the English language.
-Kathy still mourns her son's death.
This sentence refers to what Kathy does on a regular basis, a habit.
It doesn't refer to any specific act of mourning, but the fact that she mourns in general. Maybe she mourns every day, maybe only once a year.
It describes something in the present history of her life.
-Kathy is still mourning her son's death.
This sentence refers to what Kathy is doing now. Or if she is not doing it exactly at this moment, it refers to what she did in the recent past, and will continue to do into the near future.
It describes what Kathy is doing in real time, in her real life.