Sound / Tone vs. Volume
The basic tone changes because our perception of sound changes with volume. See: Equal Loudness

What are you amplifying
Even if you had the perfect microphone(s) placed in front of a great instrument - people don't put their ear(s) where you place the microphone(s)

To my ears, the sound of an Acoustic Guitar varies greatly depending on the listening position. So what is the accurate sound of an Acoustic Guitar? When playing amplified, I want a great sound, but I'm not bound to the sense that it must be "the sound of my Guitar unamplified, only louder".

Dynamic Range
As you noted above you have increased dynamic range if you choose to use it. You can do this with a microphone, but I find that I except for very quiet live environments, I get more dynamic range with a pickup than a microphone. But compared to running without amplification, with either one - pickup or microphone I have much greater usable dynamic range.

Note: If you use an L1® you have even greater dynamic range available to you compared to most conventional sound reinforcement rigs.

Compared do to conventional (point source) systems: You do not need to be as loud at the loudspeaker (Cylindrical Radiator®) to reach the back of a room, so you can play softer. You can still play louder if you want but this basically gives you a wider volume range in which to work. This is one of those interesting outcomes of using the Cylindrical Radiator® — wider dynamic range. I find the same concept applies when talking about an amplified versus unamplified acoustic instruments.

Greater range of sounds
I can do things when amplified that would simply not be audible without amplification.

This cuts both ways. Amplification brings out things I'd usually rather not hear like squeaky strings and thunks and unintentional thuds. But do any of those things intentionally, musically, and you've got a greater repertoire of sounds to bring to the mix.

Technique
When I play without amplification - playing louder means that I have to change the way I play a piece. I may strum instead of finger-picking. Or I my pick harder. This changes the sound. For good or bad, playing harder to be louder changes the nature of the sound. With amplification I can turn up the volume on the amplifier and not have to change my approach to playing.

Sustain
I have more sustain when playing amplified, and more so with pickups than with microphones. This sustain changes the way I play. My natural tendency is to fewer notes when playing amplified. The reverse is also true - I tend to play more notes when playing acoustically. I have to keep that reined in unless the song actually calls for it.

Please join the discussion that sparked these thoughts.  Microphones vs. Pickups