People always confuse sympathy with empathy. That's the reason why a lot of think that we are showing empathy when we are not. In fact, being empathetic is difficult. Very few times in our life are we truly empathetic
Compassion would be what I think is the difficult thing. Whether you fully understand what someone else is going through or not, you have an intention to help them feel better. To actually act in a compassionate way you need to first feel that others are deserving of your compassion and that can be a very tough thing to develop for anyone but those who are the very closest to you.
One side of this is just shifting of meaning over time. As this article points out, Adam Smith's Theory of Moral Sentiments (1759) gives the term "sympathy" a very central role, but his meaning is much closer to what we now call empathy.