The difference lies on an emotional level.
You may see a person experiencing hard times and emotionally you may feel sorrow or pity for them and the plight in which they find themselves. You can 'see' how tough things are for them but you don't actually 'know' what it is truly like.
With empathy you are getting an idea of the emotions of that same person. You are feeling some of the same discomfort and hardship. Maybe it's because you have been in that position before and remember what it was like and how it affected you emotionally. Or perhaps you have asked the person what it is they are experiencing and you are imagining yourself in that position based on their description.
The diagram below may help explain the difference. In the first picture the figure above the hole has sympathy for the person that is stuck in it. He doesn't care how the person feels that is trapped but he has sympathy for his predicament. He doesn't need to have experienced being in the hole to know that he's glad it's not him.
In the second diagram the figure can feel empathy because he is getting first hand knowledge of exactly how it feels and what it's like to be stuck down there. Although his experience of the situation will not be identical to that of the poor fellow in the hole, he has an emotional link that extends past sympathy which allows him to have a far better idea of what the situation is really like.
Sympathy seems to be a quality that is taught. Children will be encouraged to be sympathetic towards those less fortunate than themselves and this develops into a positive personality trait that, on the whole, benefits the individual in his day to day life. Surely it's better to be a person that has the ability to feel sorry for some tragedy that may have occurred than to not care at all and be completely unsympathetic about things.
Empathy, in my opinion, may not be so straightforward. I believe that empathy can be taught 'up to a point'. I tend to think that empathy is more of a 'skill' and, as with most skills, some are far better at it. But, again, in common with other skills, some just can't do it.
Anyone can be taught to kick a ball and then play football. Some will be clearly better at it than others because they have 'something extra'. A gift. An ability to be brilliant without really knowing themselves how. Then, as they practice more and more, they become far superior. Two people given the same instruction, shown the same skills and offered the same amount of time to practice will not turn out the same if one of them was born with that extra indefinable quality. And there will be some people that, no matter how much they try, just can't do it.
You can be told how to offer empathy. How you should react if you want to be seen as empathic. You can even be taught some empathic responses. But that doesn't mean you have empathy just because you have this knowledge. Empathy is an emotion. Empathic people are the ones that have learned to tap into that emotion. I think everyone has it in them to be empathic, unless you have some sort of medical reason why that part of your make up is diminished or absent, but it's how effectively you can tap into that part of yourself that makes the difference.
Have you ever watched a film and cried at the end when the main character finally gets what they've been striving for? That's empathy. But not everyone in the cinema cried. For some it will be because they didn't care about the character the same way you did. But for others it will simply be that they are not as empathic as you. They identified with the character, understood the plot and appreciated the happy ending, but they were not affected on the same emotional level.
Empathy is not always a convenient quality to have, especially if it means you're crying at the end of Toy Story in front of hundreds of adults and children. But if I had the choice of having it or not then I will take it, every time.