I'm still a relative newcomer in my conversion to tubes. My first, and likely last "real" tube amp is my ever delightful Yaqin MC-10L II.
I've owned a lot of solid state stuff, and most of it was very good, and there were a few "amazing" pieces as well that really knocked my socks off. However, the vast, vast majority of solid-state amps, integrateds, and receivers that I have owned cannot hold a candle to this amp, or for the better part of the tube stuff I've heard.
I personally don't buy into the solid-state vs. tube power idea- watts are watts, and 20 solid state watts is the same as 20 tube watts. What tubes have going for them, though, is the innate soft-clipping ability that helps to keep the waveform of the signal from actually being "clipped". Instead of hitting the voltage rails hard for all they've got and cutting off the top of the sine wave (as does a solid state amp), a tube amp more or less just ceases to output any more power as the limits are approached. This does give the benefit of headroom, and lower distortion when reaching the limits of the circuit.
However, I do subscribe to the superior midrange theory. Boy Wonder and I were listening to my system, and I switched back and forth between a very good fully solid state setup and the all-tube Yaqin. Indeed it was hard to put a finger on it, but for some reason the tube amp just sounded "right". After some further listening, I came to the conclusion that it was the vocal and midrange where the magic was happening. Tubes just do something to that range that is so right. So easy, clean, detailed, and nearly holographic. With a solid state setup, you might be able to choose two of those, and usually it will be the "clean, detailed" part.
There can be a certain synergy in the bass regions between a good tube amp and a speaker. This synergy is not always present, and can sometimes lead to either bloated and indistinct bass or thin and wavering. However, when this synergy is in play, the results are at least as good as a solid state amp, and, in my opinion, better. A solid state amp with a higher dampening factor (those with output transformers a la McIntosh and Sansui excepted) will generally be able to control woofer movement more precisely, meaning that it is "high fidelity" in a stricter sense. However, a tube amp, with good output transformers, can sometimes induce a more natural bass decay without smudging the tone. Even still, the DF of many tube amps is at least 10, and some are higher, which is sufficient for most speakers.
The high-end on a tube amp is another special place, where there can be some real magic at work. It should stand for something that many manufacturers of solid state amps strive for a "tube-like" quality to the higher frequencies, generally well regarded for fluid, natural, airy, non-fatiguing, high-resolution sound.
Now that is not to say that tubes are the end-all/be-all. Indeed, I am compelled to keep a solid state system around for when a speaker is not, shall we say, synergistically paired with the Yaqin. A tube amp, generally, will be more sensitive to variations in speaker impedance and how it reacts than a solid state amp will. A speaker that sounds bloated or thin on a tube amp can sound perfectly fine on a good solid state amp. Really current hungry speakers, such as Infinity, will almost certainly need a solid state amp to perform to spec, even though a tube amp may have an appropriate 2 or 4 ohm output tap. Tube amps are also demanding in terms of maintenance, and replacing worn tubes can be costly. Indeed, a matched quad of output tubes (for your typical EL34/6L6/KT88, whatever) for a common PP tube amp can exceed $100. For heavy users, this is almost a yearly charge. Vintage tube amps, particularly those of breeding and high pedigree can be very costly to obtain let alone properly restore, making the cost of entry high for those without a lot of disposable income. This can be doubly a turn-off if you look at an entry level tube amp for $500-$1000, and getting you 20-60WPC, and you see that the same money buys you perhaps a solid 200-250WPC in solid state. Speaking of power, I've found that there are times when sometimes brute-force power and panache is called for and even 50 "tube watts" just cannot compete with a solid 200+ watts for dynamic range and volume.