Some tubes were specifically designed to be used in PP, and as such, certain trade-offs were made in the distortion spectrum inherent in the tube.
One example was the 6L6, which was the first common beam-power tube- and the original fundamental design on which most subsequent beam-power tubes were generally based on. The design of these tubes used one property of push-pull amplifiers to its advantage- their inherent cancellation of even-order distortion (as a PP amp is inherently symmetric, if the tubes are equal in amplitude- as even-order distortion is inherently due to asymmetry between the top and bottom halves of a waveform, that means that when these asymmetries are cancelled, even harmonics are mostly cancelled out).
As such, odd-order distortion was reduced every practical way, in determining the geometry inside the tube- even if those practices resulted in increased even-order distortion. Then, when the tubes are used in PP, the even-order components cancelled out- leaving less overall harmonic distortion than would have been possible otherwise.
This is why 6L6s (and their little brothers, 6V6s), in particular, sound so different in SE than in PP- the increased even-order distortion of the tube gives a warm, wooly, somewhat undefined, but generally "pleasant", but not very accurate sound, in SE. OTOH, in PP, biased correctly, and run correctly in their loading, they can be EXTREMELY revealing and clean.
IMHO, because of this, they sound MUCH better, overall, when used as designed- in PP... and almost any beam-power tube will also exhibit this same behavior to some degree...
Now, pure triodes and pentodes, depending on design, may sound much more similar in SE and PP... it all depends on how the distortion spectrum is designed...
Regards,
Gordon.