I think that "comprender", is related to personal feelings, as David wrote in his example of being fired, the person has made an emotional link between his friend and his past, and he feels empathy about the bad news.

Hello Kaori, For me, both words are complete synonyms, when they refer to understand. So, I don't think there is a proper rule of thumb. However, sometimes "comprender" implies a deeper understanding than "entender", in other words, "entender" is a bit more superficial understanding as compared with "comprender". An example: Perdí mi trabajo, ¿me entiendes? —Sí, te entiendo. (I understand that you're between jobs, you can't afford the apartment now, etc) Perdí mi trabajo, ¿me comprendes? —Sí, te comprendo. (I understand your problem, I've been there. So, I know what you're going through) Por cierto, de cualquiera de las dos formas un hablante nativo entenderá lo que quieres decir. ¡Un saludo!