What contributes most to a change in how an amplifier sounds?

Power supply, quality of components in critical places, feedback, bias and even chassis design and quality. 

In a single-ended, no-feedback tube amplifier, it largely comes down to the output transformer.  Nothing else can make up for a poor OPT.  In a push-pull amp there's a bit more leeway because the OPT doesn't have to handle a large amount of standing current due to the cancellation.  Add global feedback and you can turn a pig's ear into a decent silk purse, as the saying goes.  But any tube amp is still going to benefit from quality iron.  As others have said, the power supply, the circuit design and other components can all make a difference.  Oh, and the power supply time three, as perkri says. ;-)

@thecarpathian 

😂😂😂👍🤦‍♂️ CLASSIC 

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I would agree with @helomech on distortion signature. When people describe sound attributes of a power amp, it's likely the sonic signature the manufacturer was designing for.