Piezo pickups are super high impedance, so they generally require a buffering preamp before the regular preamp in your amplifier otherwise they will have odd frequency response, often very shrill and trebly. Usually that preamp is inside the instrument, making it active and low-impedance by the time it gets to your amplifier. Having the preamp in there usually leads to a very high output level after the buffering amp. That's why you hear that piezos have both high and low output - with the internal preamp it's high output and without it would be low.
Regular passive magnetic pickups are high impedance. Amps are set up to expect this kind of output, so the level and frequency response are acceptable, but you may experience high frequency losses with a long cable (or a poor cable with high capacitance). You'll generally also find that the volume and tone controls interact somewhat (often turning down the volume causes a loss of highs).
An active instrument has regular magnetic pickups and an onboard buffering preamp which eliminates the tonal effects from the cable and can boost the signal to a much higher level. The higher output level raises your signal above the noise in the environment, so they often seem quieter. There's no more interaction between volume and tone and you can have certain features like blend controls and EQ that boosts as well as cuts. Finally, because you know that output level depends on the preamp, not the design of the pickup, you can tweak the pickup so it has less magnetic pull or wider frequency response.
You almost never will gain any benefit from an outboard preamp with a low-impedance instrument since you've already got that preamp internally. You can gain some benefits of active systems by adding an external preamp to a high-impedance system (this will solve treble loss over a long cable). You almost always will want to use a preamp with a piezo pickup, but chances are good that you already have one in most modern instruments.
David Fung
