First of all, I want to dispell the erroneous notion that solid state doesn't distort, or that it is by definition, better than tubed amps. Nothing could be further from the truth.
As Ralf says, tube distortion tends to sound better than solid state distortion. But either can/will distort, and that certainly doesn't make one *always* better than the other.
Actually, I think that there are other important aspects to the design of an amp circuit that make it sound better, and these can be aspects that can be applied to either tubed or solid state amps.
One aspect is simplicity: low feedback, Class A or high-bias A/B operation
High quality, low noise *power supply* -- this is absolutely key! Because an amp is essentially a power supply that is modulated by the input signal!
I own and listen to great examples of both:
Audible Illusions Modulus 2c preamp -- simply the best that I have ever heard. It has a single stage line amp, with a stunningly good single stage phono amp -- four 6DJ8/4922 tubes run as high gain triodes, or at least they are Class A triodes, AFAIK.
B&K ST-140 power amp (the 105watt/ch version). I suspect the bias current is set fairly high on mine. It has three gain stages per channel (AFAIK), and none of them are driven with parallel transistors.