I tend to think of electric amps being almost an instrument unto themselves. They color the tone based on the design goals for that amp. That's why all the discussions about tube vs. solid state, tube types, gain circuits, speaker types, cabinet construction etc. etc. To the point of fanaticism. Discussions talk about vintage tone, modern tone, overdriven tone . . . which are different than a pure reproduction of the output of the guitar. Obviously the pickups impact this as well but it is a whole category unto itself.
Acoustic amps are more like PAs (as pointed out above). Their major (but not exclusive goal) is to reproduce the sound that they receive subject to the space they amplify. They certainly produce a broader range of frequencies. You hear lots of discussion about favored brands of acoustic amps but they are all pointed at the best reproduction of the acoustic sound.
It doesn't mean that an electric amp, with proper EQ, can't get closer to that goal, but they weren't designed for it, and side-by-side produce very different sounds than an acoustic amp with the same input.
Some amps are dual electric and acoustic. A main characteristic I've noticed is that they have tweeters or horns to get more high frequency sounds. I'm not sure if they are cut out on the electric side. Some folks here may know more.
With more and more modeling amps out there, I wonder if they are really closer to acoustic amps but with lots of front end effects and modeling going to a pretty generic speaker cabinet. Not sure. Katana being an example . . . many many more.
Eric
K65ce
Eric