With sympathy, the helper:
Helps within his/her comfort zone
Makes a cursory judgment of the person's needs
Often will get upset when it is explicitly revealed that their help is misguided or unwelcome (after a long buildup)
May feel as if he or she is the 'mentor', or the 'superior'
With empathy, the helper:
Relates to the person on a personal basis
Forms a deep emotional bond with the person on many levels
Learns to see the situation from the person's perspective
Sees the person more as an equal, and 'walks in their shoes'.
The two should not be confused. More often than not, sympathy is the form of 'caring' that is given to those in need, and can be quite misguided, especially in dire situations that most are not used to dealing with. Most therapists, teachers, and unfortunately parents will often give this kind of 'help' to a troubled or very upset person. However, if they were willing to step outside their comfort zones, they could learn how to relate effectively.
Despite these huge differences, many indeed DO confuse the two.