There is no question that amplifiers do sound different. One of the most offputting kinds of sound artifact is sibilance on vocals, and whilst you might think it results from too much snake oil being put in, it actually results from a tendency to parasitic oscillations when excited at a critical frequency. It is hard to measure such things on the test bench, and they don't show up in specifications.
It is often described as the sound being 'too bright' and is a reason for preferring valves over solid state.
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This is another subject: the difference between a tube amplifier and a transistor amplifier.
A transistor amplifier behaves like a voltage source. Because of its high feedback rate, it maintains its output voltage equal to that of the input signal.
It does not control the current that passes through the speakers, it imposes the voltage on their terminals.
We all know that if we apply for example a square wave to a partially inductive load, the current will not be a square wave.
But the displacement of the membrane tends to follow the current's value.
There is therefore a weakness of the transistor amplifier on this point.
The tube amplifier behaves more like a current source, in particular because of its low rate of feedback and its high internal impedance. (low damping factor)
It is an essential difference between these two technologies that explains the difference in sound between the two and it is perfectly measurable, by measuring the internal resistance of the amplifier and its damping factor.