They are different things.
Closely related I'll give you, but one doesn't lead to the other. You can have empathy without sympathy just as you can have sympathy without empathy.
Here's the difference.
Sympathy is just feeling bad for someone who is having a hard time. You don't have to understand what they are going through to have sympathy for them. For instance, I have no idea what it's like you lose my mother. If you lose yours, I will have sympathy for what you're going through.
Empathy is understanding what someone is going through. You don't have to also feel bad for the person. I know what it's like to run late, but if it's your own fault you're late, I have no sympathy for you. I know what it's like to be poor, that doesn't mean I'm going be sympathetic when you vote against your best interest and mine.
Of course those two things are combined all the time. I know what it's like to lose a beloved pet and I will feel bad for you if it happens to you.
We can also never completely understand what someone else is going through. Everyone's experiences are different. The circumstances that led to whatever is going on with a person are different, as well as our perspective. That doesn't mean we shouldn't try. And when we do try we shouldn't half ass it. Trying to think of what you would feel like if something happened to you does no good if we forget to add all the circumstances to one event. Just because you think someone is overreacting doesn't mean they are. What else is going on? Is this incident the latest in a long line of small things? All good questions to ask, either the person you're trying to understand or just as a prompt to widen your perceptions.
Finally, being empathetic or sympathetic doesn't mean we need to carry other people's burdens. Most people just want a little consideration. We all have enough of our own burdens to carry.