It's the cello effect of a TBird, I'm in love with that myself. Compared to it, a Fender P, good as it might sound, always has something utilitarian.
A Bicentennial TBird is not really ideal for hard or heavy rock IMHO, the setting that Chris describes is ironically more its element. Both 60ies and modern day TBirds do better in that department. I know, I know, Pete Way of UFO, but it's a little known fact that the first three UFO albums with Schenker (Phenomena, Force It + No Heavy Petting) did not feature his TBirds at all, but only Leo Lyons' (= producer of the first three albums) Woodstock-worn Jazz Bass "because the Thunderbird didn't record very well".
And interestingly, Martin Turner of Wishbone Ash has said while he is largely identified with the TBird and its his favorite bass to play live, he's done a lot of WA recordings without it, using Rics or Fenders.