Q: Why is a different amplifier needed for a bass guitar, acoustic guitar, and an electric guitar?

A: Good question. The answer is: because the purposes of those three amplifiers are very different from each other. All amps make the sound louder, but the other things they do are different.

For an acoustic guitar, the purpose of the amp is to make the sound louder, and that is all. You do NOT want to change the sound in any way. The best amp for an acoustic guitar is the amp that changes the sound the least. For this reason, you really don’t need an acoustic guitar amp to perform. Just plug straight into the PA system and you’ll sound great. That’s because a PA system has exactly the same objective as an acoustic guitar amp: make it louder without changing the sound. So, if you plan to perform in places where there is no PA system, then just buy a PA system instead of an acoustic guitar amp! (For example, Sam’s Club generally has an ION brand 500W powered speaker with a 15″ woofer and horn with a tripod stand, two inputs, simple EQ, reverb effect, wheels, and airport style extending handle for around $200. It’s not the best PA in the world, but it’s cheaper and far more powerful than an acoustic guitar amp, and you can plug in a microphone, too. It even has bluetooth to play music from your phone.)

In contrast, for an electric guitar, the primary purpose of the amp is to CHANGE the sound. If you plug an electric guitar into a PA system (or acoustic guitar amp), it will sound HORRIBLE. That’s because electric guitars sound like crap until you change the sound. The tube circuits (or digital modeling of tube circuits) in an electric guitar amp add harmonics, increase sustain, compress the sound, add distortion, and shape the frequency response in very sophisticated and amazing ways. In fact, the amp is arguably as important as the guitar, because differences between various electric guitar amps actually have as much effect on the electric guitar sound as do differences between various brands and models of electric guitars.
For a bass guitar, the primary purpose of the amp is to make the sound really, really loud. Yes a good bass amp and cabinet does improve the sound of a bass somewhat, but the main difference for a bass amp is it takes FAR more power to make low frequencies loud than it takes to make midrange frequencies loud. Whereas a 50–100W amp is generally plenty for an electric guitar, 400–1000W amps are the norm for bass guitars.  And, of course, the speakers and cabinet are VERY different from those of guitar amps.

For a bass guitar, the primary purpose of the amp is to make the sound really, really loud. Yes a good bass amp and cabinet does improve the sound of a bass somewhat, but the main difference for a bass amp is it takes FAR more power to make low frequencies loud than it takes to make midrange frequencies loud. Whereas a 50–100W amp is generally plenty for an electric guitar, 400–1000W amps are the norm for bass guitars.  And, of course, the speakers and cabinet are VERY different from those of guitar amps.

Follow-UP Comment:

I remember when I was 14 in the 1980s and got my first cheap guitar amp and wondered why I sounded like crap. “This doesn't sound like all my guitar heroes” I thought…. It was a transistor amp designed to make an instrument louder rather than sound ‘better’.

Response:

You are not alone.  Most of the amps of the 1980s were complete garbage.  I played with a guy back then that had a fairly expensive Yamaha amp.  He spent all sorts of money on pedals and he even bought a Tom Scholz “Rockman”, and it still sounded like crap.  It was loud, but it had no musicality.  It was sad but common that you could buy a brand name amp that bore no resemblance other than its looks to the amps of those same brands being used by all the famous bands.  Solid state was supposed to be so cool.  And it was, for PA systems.  Not for guitar amps.  All those old solid state amps had “gain” knobs that would overdrive the transistors.  When tubes are overdriven, magic happens.  But when transistors are overdriven it sounds “like crap” as you said.  It wasn’t until 2nd generation digital modeling amps came out about five years ago that you could buy an inexpensive non-tube amp that sounded decent.