The biggest issue with transistors is that they don't like operating "in between". Transistors are basically an electronic switch, and they either let power through, or they block power. The challenge to making transistors useful came from making them operate in a totally variable analogue way, where one could VARY the power as easy as they could turn it on and off. This is what makes transistors suitable for audiovisual devices like TV's and radios. The problem is that to regulate the power in that "middle" state that's not quite on and not quite off means the transistor operates like a resistor...and that makes them generate heat.
The second part of the challenge was making them operate in a linear fashion. I remember learning all the equasions in electronics engineering classes that described the condustive properties of power applied to the base versus the collector. Most follow curves, either convex or concave. The better the semiconductor, the more linear the response, and the less distortion at the output that you hear at your ears. Tubes, however are very linear in their response between grid voltage applied versus plate output. When the output device (be it tube or transistor) is fully cut on, or fully cut off while reproducing a sound way, you get what is known as clipping. This is because if you were to look at the sound wave on a scope, you would see that the top and bottoms of the wave get squared off just like someone took a pair of sizzors and cut them off at a specific height.
The dirty trick that many solid-state amplifier manufacturers used to measure wattage however was operating their amplifier's transistors at maximum conduction. The obvious problem with this is that you will never achieve this while listening to music, and even if for some reason you did, you would not even be able to tell that it was music coming out of your speakers due to the clipping distortion...that is if your speakers survived that severe clipping distortion!!! Luckily, regulations in advertising have come about, and created RMS power measurements that are performed with a sine wave, and at what distortion ratio the power was achieved at.
Tubes, OTOH, are the exact OPPOSITE. They like to operate in the "in between" states, and as you approach full cutoff, or full conduction, their response becomes non linear. This is good though, because music is ever-changing, and doesn't get near these zones. If someone is pushing a tube amp near it's output limits, the clipping is much less harsh, as it sort of rounds out smoother when clipping. In many instances, a tube amp can be operated closer to it's power limitations than a solid state amp because of this fact...the distortion is simply more bearable to the ear near their maximum potiental.
One of the beauties of a tube amp too is not the tubes themselves, it's the output transformers. Like Hydraulique was mentioning, speakers don't have the same impedience across their entire frequency response. This issue however is absorbed by the output transformer, and the tubes don't see this, so frequency response is more even with less than perfect speakers. High efficiency speakers tend to sacrifice uniform frequency responce for their efficiency, but this was not a big issue with the older tube amps because of those transformers. The transformers too, were one of the reasons why tube amps were limited in their output power. Many classic tube amplifiers, especially McIntosh, like Phil mention got their reputation NOT because of them being tubes, but because of the superior build quality of their output transformers. Building an output transformer that will be resonant from 20hz all the way to 20,000 hz is a serious engineering challenge. Output transformers too, can also absorb clipping distortion, becasue the magnetic flux built up in them will fill in when clipping occurs. A few high end solid state amplifiers have been built with output transformers for the exact reasons stated here. This is a rare feature however because output transformers are expensive, and really aren't necessary on a solid state amp.
