The sonic term 'warm' comes 2nd harmonics generated by a tube run without feedback, running a BJT without feedback is denoted by terms like 'harsh', 'unpleasant' and 'awful'
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Tubes win however not just on an intrinsically musical distortion but also that often we use a smaller part of their transfer curve due to the high voltages the anodes sit at.
The other really neat things for DIY is that you don't have to buy heatsinks and they are difficult to kill, although the voltages can kill you and generally you'll need to find transformers for power amps - so it's swings and roundabouts I guess.
The biggest flaw of transistors in my view is their small size, tempting designers to scatter them liberally around the circuit which distracts from the essential purpose of the amp, the cost of a tube in terms of size and power means they tend to remain smaller in number! MOSFETs do make far better followers though, I'll give them that.