Some people are better at empathy than compassion, so they tend to focus on themselves when they encounter distress. There are two ways to respond to suffering:
Compassion is tied to the present moment. A compassionate person resonates with the distress in front of them, rather than the events that brought the suffering about. Compassion is an immediate, uncomfortable sensory experience. Because it's embodied, it's often tied to action, but is frequently without words. When someone dies, compassionate people are often silent. We're busy resonating.
Empathy is tied to the events that caused the suffering. The listener thinks about how they would respond if similar events were to happen to them. Because empathy is mental and not embodied, it keeps a little safe distance between the observer and the sufferer. While compassionate people fall silent, empathy gets people relating.
As we empathize, we focus attention on ourselves. If the event that is creating someone else's suffering is trivial (in our estimation) we feel contempt for the "whiner". If the event is too enormous for us to relate to, we respond with pity. In the sweet spot between these two extremes we "can relate". Because empathy immerses us in our own experiences, many people talk about themselves:
"Sorry about your cancer. When I had cancer blahblahblah."
At its most refined, empathy gets us thinking about the golden rule : "If I was feeling this, how would I like other's to respond." Still, to the sufferer, this sort of self-conscious "acting" is less comforting than a few moments of genuine compassion.