This concept is an analogy and gives a memorable name for the instrument.  As I understand these terms in psychology, sympathy means "I feel for you," whereas empathy means "I feel with you."  There is a distance with sympathy and more of an intertwining with empathy.

While, of course, strings can not feel in way humans do, this analogy does at least point to the subtle but important difference between sympathetic strings which we hear, for example on a sitar, and the empath bridge system.  Sympathetic strings are those which are attached to the same instrument body as their source, usually running under the played strings or fretboard.  When a played string shares one or more harmonics (modes of vibration) with a sympathetic string, it is excited and vibrates in one of its vibrational modes for the shared harmonic(s).  It can't help it. 

 

The same phenomenon can occur from a separate instrument or resonating body that we will call sympathetic resonance.  For example, the strings inside a piano, when someone depresses the sustain pedal, with resonate sympathetically when other strings are played or when a person yells or sings into it.  Only the shared harmonics resonate.  In the case of most pianos, which are not tuned to Just Intonation, the harmonics may be activated by pitches that are close enough to excite their harmonics.  Maybe you've played a loud distorted guitar and when you stopped you heard an acoustic guitar across the room ringing.  Or that pesky snare drum buzzing when you play bass? Sympathies.

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In these cases, the sympathetic strings/resonators can only be actuated or not by the source, and once they are, they simply ring out.  The empath bridge system described below has more direct control/interaction with the strings so that if the shared harmonic in the source stops, the strings stop too.  The empath bridge starts and stops the strings and allows for changing pitches quickly. 
 

Sympathetic strings are actuated by the source (usually other played strings) creating a feedback loop which extends the resonance.  In contrast, the Empath instruments work passively with the audio source so that it actuates the strings without feeding into the soundboard.  The pickups can send the string-only signal to an amp without a feedback loop.