I first became aware of the importance of the difference between empathy and sympathy when I used a professional jury consultant in a case. He cautioned against picking jurors that were too demographically similar to the client. For example, if the client is a 30 something female, your ideal juror would probably be an older person who would view the client as a daughter. You don't want to pick another 30 something female juror because that person would empathize in a negative way--she would tend to be too judgmental. And zeke, the consultant said statistics show that this judgmental empathy is the most severe in the 25-40 female demographic. Thought you would want to know that.
In my own practice, I always thought that I could be a better advocate for my client by being a little distant. I've mentioned before that I was hit by a drunk driver and had many non-life threatening injuries. I was in a position to put myself in the position of my client. I guarded against that. It's hard enough maintaining objectivity in a case without putting your self in the client's shoes. Clients usually need to know that they come across as convincing and sympathetic to an objective and outside observer. They don't always get that with an excessively empathetic person.
There are a lot of . . . .ahem . . . . .nuances to this point. I think it is important to not conflate the two ideas.