I often feel empathy without feeling any compassion. Compassion for the despicable is rarer for me than empathy. I only feel both empathy and compassion when it really is "there but for the grace of God". Like Andrea Yates. Having dealt with mental illness/dealt with mentally ill people, I could both empathize based on my own experience and feel compassion. But for Saddam Hussein, I had empathy (I know what it's like to be egomaniacal and have a messed up family, etc.) but no compassion. Compassion is for those who are human.
I do believe some people are more human than others. "Human" is probably not the most precise word, but it captures what I'm saying. So for me, Andrea Yates is fully human and warrants compassion. Whereas Susan Smith was less human and did not deserve what compassion she got. I could empathize with Susan Smith, I felt what she was feeling (desperation, fear of rejection, lust, pressure). But I have no compassion for her.
As for pity, it's not something I experience very often and when I do, I dismiss it. It feels fake for me to express pity, because I don't really feel it, not deeply. If I did, I feel like I'd do something about it.