A major problem with comparing two amps, one burned in and the other not, is that you have to be certain that both were identical to begin with. Even with two units of the same make you have variation between parts. They're usually matched for each unit, but not matched across every unit ever made.
To put this a different way, two brand new units of the same model could very well sound different, regardless of the amount of time they've "burned-in." Unless you know the two units were identical to begin with, a comparison is useless.
I can offer a different perspective on this than many. Before I got into Hi-Fi, I spent many years restoring old radios. I've gone though at least 100 of them over the years and have had the chance to listen to 50 or 60 year old sets. My oldest radio was made around 1926. Without question, parts degrade over time. This is a function of heat and time of use. Most components sound best when new or shortly after. You'd be shocked at the difference in sound when you replace old caps with new ones. Time and heat take a toll on every component.
This is why I'm disinclined to believe that time (and the associated heat) somehow improve things. Capacitors form up within seconds (there are formulas for this, even) and degrade from there. If you want to call that an "improvement," then that's your taste. From a technical standpoint, components degrade and drift in value from the minute you start using them. Browse the manufacturers' spec sheets at Digikey and Mouser to find the MTBF (mean time before failure) for the components you have.
The main reason why components sound "different" when warmed up and usex for awhile is because time and heat cause them to change value after awhile. Again, if you think they sound "better" after this, that's your taste. Technically, they've degraded and are not as close to spec (and what the circuit calls for, by the way) as they were when new.
Which is why I don't put muck stock in "burn in" and also why I make sure to turn off components when not in use. I know the clock is ticking every minute I use something and I'm not eager to warm up the iron to repair something prematurely.