The term 'Acoustic Amplifier' is essentially a contradiction in terms as once an acoustic instrument is connected to an amplifier it is effectively no longer acoustic, but electric or electro-acoustic.
Acoustic amplifiers are amplifiers (usually in a combo configuration) that have been designed and built to amplify the sound of an acoustic guitar. In the early days of folk music, most folk clubs had a small and intimate gathering of no more than a few dozen spectators who would come to listen to acoustic guitar players and therefore amplification was not a requirement.
However, with the growth of acoustic music, bigger and bigger venues were required as the public warmed to the new folk artists. This in turn required the acoustic instruments to be amplified in some way to accommodate the bigger venues.
Acoustic amplifiers meet this requirement by allowing electro-acoustic instruments to connect to a source of amplification that has been specifically designed for this purpose. The combination speaker and amplifier also make the unit more transportable.