Sympathy is the broadest understanding of the word. Empathy is more an ability to relate to something, to be able to put yourself in that position.
It's complicated either way.
"Synonyms:
1. concord, understanding, rapport, affinity.Sympathy, compassion, pity, empathy all denote the tendency, practice, or capacity to share in the feelings of others, especially their distress, sorrow, or unfulfilled desires. Sympathy is the broadest of these terms, signifying a general kinship with another's feelings, no matter of what kind: in sympathy with her yearning for peace and freedom; to extend sympathy to the bereaved. Compassion implies a deep sympathy for the sorrows or troubles of another coupled to a powerful urge to alleviate the pain or distress or to remove its source: to show compassion for homeless refugees. Pity usually suggests a kindly, but sometimes condescending, sorrow aroused by the suffering or ill fortune of others, often leading to a show of mercy: tears of pity for war casualties; to have pity on a thief driven by hunger. Empathy most often refers to a vicarious participation in the emotions, ideas, or opinions of others, the ability to imagine oneself in the condition or predicament of another: empathy with those striving to improve their lives; to feel empathy with Hamlet as one watches the play."