Oh boy... I actually addressed tube vs. transistor (BJT and FET) distortion in another thread, but I am not sure which one...
1. The intrinsic distortion series for a bipolar transistor is exponential. It is factorial for a tube. That means the distortion products are actually lower for a tube than a transistor, all else being equal. An ideal FET's distortion series (Taylor expansion) ends at the second order -- no higher terms, making it the winner. Of course, in the real world, other things ensure there's plenty of distortion to go around...
2. Tube circuits are most often single-ended with low feedback factors. Transistor circuits tend to be differential (internally if not at the input) with higher feedback. This means a tube circuit tends to exhibit even-order harmonics, and a transistor circuit odd harmonics though at a lower level.
a) There are lots of tests and papers saying we (humans) find even-order harmonics more "pleasing" than odd-order. A rounded sinusoidal pulse has higher even-order terms; a square wave, higher odd-order. The rounded cosine pulse sounds "softer" than the raspy buzzing square wave.
b) Tube circuits tend to saturate "softer" than transistor circuits for a variety of reasons. They tend to round off peaks before hard clipping, where transistor circuits, well, they just clip! This has led to people feeling tube circuits clip "softer" than transistor. This is not fundamentally true, but you will start to hear the distortion a little sooner in a tube circuit and turn it down, while in a transistor circuit it just wops you upside the head. (That's technical talk.)
3. The type of distortion that people hear/dislike the most is non-harmonic, e.g. intermodulation distortion (IMD). (And, I do mean ergo (e.g. -- for example) as there are other ways to get nonharmonic distortion terms.) Harmonics we can handle better than something not related (or loosely related) to the signal. Two tones generate "sidebands" at the difference frequencies and multiples:
f1 harmonics = 2f1, 3f1, etc.
f2 harmonics = 2f2, 3f2, etc. -- all multiples of the signals, bad but not too bad
IMD = (f1-f2), (f1+f2), (2f1+/-f2), (f1+/-2f2), etc. -- not multiples of the signals, worse
4. Achieving a high damping factor in a tube amp, with its output transformer, is very difficult. The actual culprit (imo) is load regulation; the relatively high and fairly complex output impedance means a tube amp does not produce a well-controlled flat response into a typical (fairly ugly) speaker load. I loved my tube amp (ARC D79) on my relatively benign Maggies; I thought it did poorly on a set of B&W 801s.
5. High feedback in a transistor amp yields impressive THD numbers and excellent IMD as well, but it also tends to cause high-frequency peaking and phase shifts that induce transient IMD (TIM) because the feedback path cannot immediately and precisely correct the output to match the input.
FWIWFM, my 0.000001 cents (microcent), IMO, blah blah blah - Don
p.s. There have been tube amps designed with differential circuits, additional cathode followers (buffers), and direct-coupled inputs and outputs. They provided distortion characteristics more similar to transistor amps, and were of course panned for it as a result.