Valves are voltage controlled current amplifiers, and ideal for audio use. They're generally extremely linear over the audio bandwidth (and beyond). unlike transistors, and so because of their 'natural' linearity, the circuits can be simple with very little negative feedback required.
Transistors are the opposite, they work within a smaller range, requiring many in a circuit, requiring in turn more complex circuits with all the attendant power supply/ground path issues. They also switch (in Class AB) which causes lots of distortion, mainly nasty gritty 3rd harmonic. To slash this back, negative feedback is applied, which gives that 'sat upon' sound.
Valves, by contrast, produce lots of distortion but it's mainly second harmonic, which is actually euphonic as taken by the human ear.
Neither system is perfect, but the benefits of valves (well done) in a circuit often outweight the pitfalls, in my opinion.
Non linear transistors were never designed for audiophile use; almost like trying to drive a car with square wheels, some would say. But fascinatingly there was one transistor designed to work just like a triode valve, giving great linearity and lots of current (hence simple circuits and low feedback). It's called the V-FET; designed by Sony and Yamaha in the early 1970s. It's back!
http://www.digital-do-main.com/product/index.html
Personally, I use the Sugden Class A amps, which are extremely low measured distortion (no switching goes on, the transistors are powered up and on all the time). It's very clear, much more so than Class AB, but still a little less 'free' sounding than valves, likely explained by the circuit's relative complexity.