You can plug an acoustic electric guitar into an electric guitar amp and get a sound, however they’re not designed for that purpose. You’ll probably find that an electric amp will add things to your sound that don’t quite work for an acoustic – certain frequencies, gain etc. In some ways, acoustic amps aren’t too dissimilar to a PA system in that they’re designed to amplify a signal, without adding too much color to it.
Acoustic amps quite often will feature an additional smaller speaker called a tweeter too – this helps deliver clear and defined higher frequencies so you get a nice, balanced sound, full of richness and clarity.