Here are 2 definitions:
Empathy - The ability to understand and share the feelings of another.
Compassion - Concerns for the sufferings of others.
Empathy and Compassion are not the same. One has to do with how you feel, the other has to do with how you act. As a manager, you strive for both, because you need both to build solid and engaging relationships.
If an employee asks to leave early to see his son's play, you can have empathy and say "I know how important it is to be there for those little moments", but then turn around and say "no". That is empathy without compassion. You will get no respect from your employee with this.
Same situation, but reverse, you say "I can't understand why that stuff matters, but sure, go ahead". The employee will go, but you will seem like a heel for not understanding why it matters and will have damaged the relationship.
The great manager has both empathy and compassion. Empathy to understand your employee and their feelings, compassion to act accordingly.
One caution, it doesn't mean that you always give the employee what they want. There may be instances where you just can't - but compassion means being transparent and understanding why they might be upset.