There are so many aspects to cover, but here are some of at-random $0.02 from a geeky music lover who used to build DIY tube amps in his youth...
First, like JohnK, I subscribe to the thoughts that, when it comes to the amplifiers, the sonic quality is essentially 100% measurable. I do NOT say the same for the loudspeaker sound, which I still believe has so many complicated aspects that even tomorrow's cutting-edge measurement systems cannot completely cover every attribute of sonic quality. This is especially true if you consider the room/speaker complex as an integrated transducer system. However, when it comes to source players and amplifiers, I do think that essentially ALL audible aspects are already covered by the physical measurement methods available today.
Second, it is true that many if not all tube amps exhibit definitive sonic signatures that are both readily measurable physically and clearly audible in well-controlled blind listening tests. The high output impedance of tube amps mentioned above is one of the major reasons for this. In other words, if you define the "ideal amplifier" as the one that does nothing to the input audio signals but faithfully amplifying them, then very few, if any, tube amps can even come close to today's solid-state amps, including fairly cheap SS amps.
However, as others have already pointed out, some people define the ideal amp very differently, and it is true that signal "aberrations" induced by some tube amps are indeed subjectively pleasing to many people. For example, single-ended triode designs tend to add a lot of even-order harmonic distortions -- basically, distortions consisting of the signals exactly one octave above the input sound (2nd order), two octaves above (4th), two octaves and fifth above (6th), three octaves above (8th), and so on... These distortions are MUCH less harmful musically than odd-order harmonic distortions, and sometimes even positively pleasing to people's ears (in contrast, the 5th, 7th order harmonics and above are "out-of-tune" as you know). I fully understand that these clearly audible sonic characters of tube amps add a lot to the enjoyment of the audiophiledom as a hobby.
Third, in my personal opinion, all commercial tube amps in today's market (except for some DIY "kits") are BADLY, GROSSLY overpriced due to this audiophilic tube frenzy of almost religious magnitude. Tube amps today contain essentially ZERO new technologies as compared with the ones made in the '60s -- except for those "frills" such as digitally controlled bias-regulation circuits etc. The manufacturing costs of some parts (including the vacuum tubes themselves) have understandably gone up a little due to the low quantities of production. Yet, as far as I can tell, no tube amp vendors could possibly justify those stratospheric product prices if not for the demands from the relatively small body of quasi-religious, cost-no-object tube fanatics. Of course, the same argument applies to many other absurdly priced "audio jewelries" -- but IMO, tube amps represent a good example of the "super-high-margin" product categories.
