The room hears the sum of all of these things, but the mic only hears what it is happening were it sits. Also, mics aren't very discriminating. At low frequencies they can't distinguish the direction that a sound is coming from, and so can pick up both the room and the instrument, as well as things like talking, bumps, and so forth. The upshot is that mics feed back more and are noisier than most other types of pickups. Their real strength is in their low-frequency sensitivity.
Soundboard transducers can be either piezo elements or magnetic motion sensors.
The magnetic type use a small magnet glued to the soundboard with a wire coil nearby. Motion of the magnet causes a current in the coil. As with magnetic string pickups and mics they are quite sensitive to low frequencies, but they don't get in the way like string pickups, and are not as noisy as mics. Piezo transducers can be either of the accelerometer or strain gauge types. Piezo materials become electrically charged when they are distorted.