You can certainly use a standard amplifier for your acoustic-electric guitar. Before the advent of acoustic amplifiers, this is exactly what acoustic artists would do. However, they were not satisfied. The effects (typically only distortion) and EQ sections were designed for electric guitars, and did not lend themselves particularly well to acoustic guitars. As a result, acoustic guitars sounded rather dull and flat. In addition, since many acoustic artists also sing, they wanted an amp that they could plug their microphone into so they would not need a separate PA system. The acoustic amplifier addresses these issues. The EQ section is tailored specifically for acoustic guitars, as are the effects (chorus, reverb, delay). Acoustic amps also have a second channel with an XLR input for a microphone. The microphone channel also has its own EQ section and effects, specifically suited for vocals.

In short, the acoustic amplifier allows the natural sound of your acoustic guitar to shine through and is a must-have for acoustic artists.