A piezo transducer is a crystal that responds to pressure (including vibration and air pressure) by creating electricity. Piezos can be quite small and accordingly can be mounted in small spaces like those found in a violin bridge. Because they respond to pressure of any kind, piezos will respond to more nuances of an instrument - body resonances and bow noise will both be picked up by a piezo. The down side is that because they pick up body resonance, they can feed back when used on an instrument with a sound hole (or in the case of a violin, an f-hole). This makes them difficult to use at high volume (on stage, for example) unless care is taken to keep them away from monitor speakers.

Magnetic coil pickups are what you see on electric guitars. A central metal core is wound with wire to create a magnetic field. When that field is broken by the vibration of a metal string, a electrical voltage is induced in the coil. Because coils don't respond to air pressure they can be used at much higher volumes before feeding back, and tend to be somewhat smoother (and less detailed) in response than piezos. This can be good or bad, depending on your application.

It's important to note that piezo transducers of the size used in violin pickups really need a preamp to sound good. This is because they are high-impedance devices, and when connected to a low-impedance amplifer they tend to sound shrill or gritty. Most of the signal available gets used trying to drive the amplifier, leaving little for full, rich tone.

Some makers partially alleviate this by connecting a 1megohm pot (variable resistor) to the output of the pickup. The pot, when at its "full," or 1megohm, setting can provide enough additonal resistance in the circuit that the amplifier gets to do more work, leaving the tone of the pickup more or less intact. But that assumes the pot is turned to its highest resistance, which in the case of a volume control means its lowest volume. Although tone is preserved, the signal at the amplifer is quite low and has to be increased to compensate. Also, a simple pot connected to the pickup alters the tone as it moves through its range. At lowest volume, the tone is warmest and fullest, but also quiet. At full volume, the pickup is again shrill (because the pot is no longer presenting additional resistance in the circuit), albeit loud. Ideally you want the tone of the violin to be consistent throughout the volume spectrum. For that, you need a preamp (also referred to as an impedance-matching preamp) that can boost the output signal of the pickup prior to main amplification.

Confusing? Yes. Obvious once you listen to a violin at different volumes through the different devices? Yes. I guess the short answer is, for best tone, get a preamp. You'll be glad you did.