Sympathy essentially implies a feeling of recognition of another's suffering while empathy is actually sharing another's suffering, if only briefly. Empathy is often characterized as the ability to "put oneself into another's shoes". So empathy is a deeper emotional experience.
Empathy develops into an unspoken understanding and mutual decision making that is unquestioned, and forms the basis of tribal community. Sympathy may be positive or negative, in the sense that it attracts a perceived quality to a perceived self identity, or it gives love and assistance to the unfortunate and needy.
I think for me as an INFJ, my Ni allows me to empathize with the person and my Fe-aux is the outlet that I offer sympathy (or contempt)...
Thru Ni I understand where the other person is coming from, how (s)he feels, what his/her reasoning and then may decide to employ Fe to externally show sympathy and offer help or criticism depending on whether I agree or disagree with him/her...
It is possible to be empathetic and not sympathetic at the same time.
For example: If a person gambles and loses all his money, you may feel empathetic and try to analyze the reason for doing so but you will not be sympathetic towards him as it is his fault entirely in losing the money.
On the other hand, you can both empathize and sympathize at the same point. If someone loses a loved one to a disease, you will feel sympathy for them and, if you have ever lost a loved one yourself, you are likely to empathize with their position.
Another example that captures the difference between empathy and sympathy: "When I think about the abuse the serial killer endured as a child, I feel empathy, however I simply cannot sympathize with the choices he made as an adult."
When one exhibits empathy a person doesn't necessarily have to agree with the conclusions being drawn by the person who they are empathizing with. For example, one may empathize with the loss of a loved one but may not agree with another person that the loss be avenged violently.
So I believe it is not entirely about Fi versus Fe...
For INFPs, Fi-dom is the reservoir of their feelings and values...If what the other person tells resonates with something within the Fi-reservoir (thru Si-tertiary perhaps?) INFP can empathize with the other person (and judge him/her since F either introverted or extraverted is a judging function?)...and then depending on the judgment perhaps may employ Ne to find ways to soothe/help the other without passing any judgment (since N either introverted or extraverted is a perceiving function?)...
So you can empathize without sympathizing but cannot sympathize without empathizing?