Imo, I think empathy (defined as the ability to understand and share the feelings of another) is more valuable than sympathy (which generally comes from a "pity" stand point). When I think I am empathizing with someones struggle - I try to recognize whether i'm actually empathizing or sympathizing. I think I am empathizing when I feel as if i've put myself in the other persons shoes and feel moderately close to what they are feeling.

I remember in high school, there was a set of twins. The parents couldn't afford to send both on a trip, so they had to pick one of the two. Long story short - the conversation led to them crying and then me crying. My teacher looked puzzled and asked why I (who had nothing to do with this conversation) was crying - and I realized it was because I empathized with them.

You're right about us not being able to truly feel beyond simulation what someone else feels, but I think empathy is when it feels just as real to you as it does to them - even if it doesn't affect you in the same way.