I believe that empathy in many cases is a prerequisite to compassion. Empathy is not only understanding how exactly that other person is feeling, and empathy is not only for sad moments, but for the moments of joy too. Person who is lacking empathy will rarely feel joy for somebody else success. While we might be compassionate for two reasons, one being emotional and another rational, empathy is more of an emotional trait of a person. That is probably why you were having a discussion with your wife about it, as women tend to accept and analyse situations more form an emotional point of view while men being more rational thinkers. And since it is a feeling it is really difficult to explain, as it also may be felt in different levels so it is hard for us to grasp what exactly the other person is meaning when talking about empathy. It's just like trying to explain what love is.
I agree on the prerequisite of one to the other. Plus dissecting these concepts in a logical perspective risks to betray the true nature and aim of these movements of the human soul. Com-passion / em-pathy ethimologically share the same ancient greek root, which is pathos, feeling.
So that only proves they have similar background. The rest is just interpretation that we people put into it. And all of us do it in a slightly way so there is a lot of space for misinterpretation. Hence discussion :)