I think I'm finally starting to understand the term "Fender clean". It's a wide-range sound with solid lows and extended highs. There's a nice piano-like attack on low notes, and a shimmering complexity to the high notes. The amp transitions into (and just as important, out of) distortion smoothly; some designs transition more or less gradually, but they're all smooth.
The extended frequency response is both a blessing and a curse. When pushed too hard, the pianistic attack can become flabbiness and the shimmer can give way to an ice-pick shrillness. But... and this is important... the extended frequency response is essential to the Fender sound. Take it away and you have an amp that's easier to play at the expense of limiting the expressive options you'd get through varying the intensity and style of your picking.
Big-bottle American output tubes seem to be another important part of the equation. 6L6s, 5881s and even 6V6s have the sound. EL-34s and (especially) EL-84s don't.
It's funny... Kevin O'Connor, author of a very useful series of books on guitar amplifier theory and construction, wrote that the Mesa/Boogie sound limits the highs and lows and biases the outputs cold, an assertion I have firmly resisted. Maybe the Rectos, sure... but the Mark series has a nice extended response. And that's true within the Mesa line. But when I put my Mark IV in the same room with my Vibro-King and '62 brown Pro, the difference was obvious. Just as Kevin wrote, the lows and highs simply weren't there on the Boogie. And the transition into and out of distortion is noticeably smoother on the Fenders, perhaps a result of their hotter output bias.