When you understand and feel another person's feelings for yourself, you have empathy. It’s often spoken of as a character attribute that people have to varying degrees. If hearing a tragic news story makes you feel almost as if the story concerns you personally, you likely have the ability to empathize. You have to be able to put yourself in their place and FEEL their emotions.
When you sympathize with someone, you have compassion for that person, but you don’t necessarily feel her feelings. If your feelings toward someone who is experiencing hardship are limited to sympathy, then you might have a sense of regret for that person’s difficulty but are not feeling her feelings as if they’re your own. Meanwhile, sympathy has broader applications that don’t necessarily have to do with one person’s feelings for another. You can sympathize with a cause, for instance -- breast cancer, or with a point of view that resonates with you.
I'm going to be blunt here... but it's close to impossible for someone to empathize with a bereaved mother unless they too have lost a child. You might be able to give sympathy or support, but to really be able to understand or know what a mother who has lost a child feels, you have to have been put, tragically, in her shoes.
I want you to know I don't want you to empathize. I want you to sympathize. I know that it's hard to imagine (and I would have never imagined) this happening to me. But what I want you to do is to support me and speak her name. I want you to "like" posts in her memory. I want you to never, EVER, have to feel the pain I feel every day.