Much of the reason tubes sound better is the spectral distribution of the distortion components which every audio component produces. Solid state amplifiers produce lower (by an order of magnitude or more) static distortion numbers, but the quantity of the distortion is not nearly as important to the ear as the spectral distribution.

Tube amplifiers produce a nearly "ideal" spectral distribution of distortion components with a strong 2nd harmonic, which masks the presence of the offensive higher-order distortion components, with the 3rd, 4th and 5th desending in level quite rapidly.

This can be seen in this photo of the distortion produced by a 1kHz signal in a single ended triode amplifier. A tube buffer would have a very similar distortion spectra.



By contrast, here is the distrotion spectra of a 1kHz tone in a high quality solid state amplifier.



Notice that the 2nd harmonic at 2kHz is almost totally missing, as are all of the musically-related even order components (the 4th, 6th, 8th harmonic etc). The presence of the 2nd harmonic is very important for good listening quality because it masks the presence of all other distortion components. Now look at the other parts of the spectrum - there is high-harmonic-order distortion all the way past 10kHz, and this is completely unmasked by the lack of a 2nd order component!

Tubes are also highly linear devices themselves and in amplifiers can be run with little, or no, global negative feedback. By contrast, solid state gear must make use of high levels of global negative feedback, otherwise the distortion would be intolerable.

There are other advantages of tubes, but these are the most important ones.