There are two main types of pickup: magnetic and piezoelectric. Electric guitars use magnetic pickups, in which small coils react to the movement of the metal strings. Magnetic pickups can also be used on steel string acoustic guitars, but won't work with nylon strings, so many acoustic guitars (and all nylon stringed ones) use piezoelectric pickups, in which a small disc of piezoelectric material picks up vibrations directly from the body of the guitar (usually somewhere on the back of the soundboard).
People using steel-string guitars sometimes like to use both, either by using a soundhole-mounted magnetic pickup in addition to a sounndboard-mounted piezo, or by using a hybrid pickup which combines the two technologies. This can be essential if you're playing in a percussive style - a traditional magnetic pickup won't react to body slaps (though there are "floating" designs around now which do).
With a magnetic pickup the choice of wood etc. will only affect the sound very, very indirectly and will probably not be discernible. With a piezo pickup you are recording the vibrations directly from the wood, so if there is a difference between different tonewoods (and that's a big if...) and if your pickup has a good enough frequency response then you should in theory be able to hear it in the output.
A good pickup can produce a pretty nice sound, but it still won't pick up all the subtleties of an acoustic guitar's sound because it's only listening to one spot on the body (or sometimes two). I have heard some very nice recordings of acoustic guitars through pickups, but in my personal experience you get a better result with a couple of good mics, or preferably just your ears.
FWIW I have a very expensive luthier-made nylon string which has a lovely rich sound but sounds completely dead through its relatively high quality pickup.