These pickups are generally high impedance devices, and thus have a limited frequency response made worse by running long lengths of cable from the guitar to an amp or board. The result is that the upper harmonics are lost. Feedback suppression is reasonably good as the pickups are not picking up vibrations directly from the top, and the partial blocking of the sound hole cuts down acoustic coupling between the guitar's air resonance and the amplified sound field. A magnetic pickup's output will sound much like the guitar it is mounted on, rather than a Strat or Les Paul, because the sound of the guitar is reflected in how the strings vibrate. Conversely, the same pickup mounted on a solid body will sound like a solid body guitar pickup.
Magnetic pickups are relatively inexpensive, convenient and moveable between instruments.
Electrodes are attached to specific surfaces of the crystal so when the crystal, packaged as a pickup, is subjected to vibration the output is sent on down the line. The main problem with piezos is that they are extremely high impedance devices. As such, the frequency response you can practically use is greatly affected by where the piezo output is sent. It takes a very high input impedance to match the high output impedance, in order to prevent severe low end roll off. Otherwise the result is that typical, honking 'cardboard' sound of guitars with piezos with no matching preamplifier.