I don’t really have a horse in this race, but thought some of you might find it interesting. From what I’m hearing, there’s no real difference in frequency spectrum per se, but there is a difference in dynamics, and since distortion “adds” harmonics, that can come across as a shift in frequency response even beyond the “louder is better” thing. If you’re hearing harmonics from one tube where the other didn’t have them (because it wasn’t distorted as much), that could be noticeable.
Pursuant to the above video (which I wholeheartedly endorse!), I've found a very basic tutorial on how tubes work. This guy is a bit more "dry" than the previous presenter, but he's absolutely spot on in giving you what you really wanted to know about tube operation. His examples are good as well, as he relates electron theory to day-to-day things around the house. Except for his short blurb about "electron theory versus conventional theory" (wherein I disagree only with his labeling of such), he's a veritable fountain of knowledge. He even has a lot of ancient tubes for display and teaching purposes, a few of which I've seen only in the classroom, never in a real-world scenario - that's impressive! Here, try this on for size:
There are a number of suggestions alongside of this video, but the ones I checked out seemed to be "lacking", for want of a better word. Still, you could (and should!) cast around for similar stuff, if you're thirsty for more on how tubes and amplifiers work.
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Rule #2: In case you don't like the life you have, see Rule #1.