The transfer function for a good-quality solid-state power amplifier is almost perfectly linear through its entire output range, until it hits the power-supply rails. When the output hits the rails, the signal is "clipped."
It's as is someone came by with a pair of scissors and just cut the tops and bottoms off your nice, pretty sine waves. In other words, the clipping introduces discontinuities in the derivatives of the signal, which introduces unwanted harmonics. For some reason, these harmonics sound ugly -- or even unnatural -- to human ears. None of the classical instruments made of strings and pipes are capable of making such jagged waveforms.
On the other hand, tube amplifiers have quite non-linear transfer functions, particularly as their otuputs get closer and closer to the rails. This is a very unwanted phenomenon in communications, sound recording, and many other situations, which is most modern amplifiers are solid-state. When you pick up a guitar, though, you're not aiming for linearity.
Since tubes more gently "round" the tops of large signals, you don't get the strong, unwanted individual harmonics. Instead, the gently rounded sinusoid spreads that unwanted harmonic energy over a large bandwidth. There's second-, third-, fourth-, and even fiftieth-harmonic energy in that sound -- it becomes a rich tapestry of tonal qualities all mixed together. Then the amplifier manufacturers provide a couple dozen knobs to turn, each gently shaping that harmonic envelope, and give musicians reason to spend rainy weekends "working on their tone."
So why does the tube amp sound better? Psychoacoustically, humans are not capable of detecting a small sound with a frequency close to that of a large sound. Futhermore, the ear's response is logarithmic -- even if a single harmonic contains half as much energy as the fundamental tone, is it not perceived by the ear as being anywhere close to "half as loud." As long as none of the harmonics compete fiercely with the fundamental, you'll probably enjoy the sound.
As a result, even a very heavily clipped (over-driven) tube amp continues to produce a pleasant sound. Heavily-clipped sounds with lots of harmonic energy all over the map may be considered "dirty," or "gritty," or "fuzzy," and are even the subject of intense envy if you're trying to sound like Jimi Hendrix.
If you drop your Marshall stack out a window and have to settle for a Crate, you'll be in for a different experience. An overdriven solid-state amp just assaults you with a very strong, very unflattering second or third harmonic nearly as loud as your fundmantal tone. Instead of creating a rich, full sound reminiscent of the warmth of sunlight's continuous spectrum, you're staring at the monochromatic ugliness of a naked fluorescent tube.
- Warren