sym·pa·thy
Pronunciation: 'sim-p&-thE
Function: noun
Inflected Form(s): plural -thies
Etymology: Latin sympathia, from Greek sympatheia, from sympathEs having common feelings, sympathetic, from syn- + pathos feelings, emotion, experience -- more at PATHOS
1 a : an affinity, association, or relationship between persons or things wherein whatever affects one similarly affects the other b : mutual or parallel susceptibility or a condition brought about by it c : unity or harmony in action or effect <every part is in complete sympathy with the scheme as a whole -- Edwin Benson>
2 a : inclination to think or feel alike : emotional or intellectual accord <in sympathy with their goals> b : feeling of loyalty : tendency to favor or support <republican sympathies>
3 a : the act or capacity of entering into or sharing the feelings or interests of another b : the feeling or mental state brought about by such sensitivity <have sympathy for the poor>
4 : the correlation existing between bodies capable of communicating their vibrational energy to one another through some medium
synonym see ATTRACTION, PITY

em·pa·thy
Pronunciation: 'em-p&-thE
Function: noun
Etymology: Greek empatheia, literally, passion, from empathEs emotional, from em- + pathos feelings, emotion -- more at PATHOS
1 : the imaginative projection of a subjective state into an object so that the object appears to be infused with it
2 : the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings, thoughts, and experience of another of either the past or present without having the feelings, thoughts, and experience fully communicated in an objectively explicit manner; also : the capacity for this

Sympathy does imply some "pity" whereas empathy suggests supportive understanding. I agree that sympathy may feed self pity.