Franck and WIdor
Hi Thomas,
Overall, I'd say that Franck was somewhat more inspired and that Widor was somewhat more influential.
Most of the "Franckists" were somewhat at the margin (e.g. Ropartz) or even outside the organ crowd (e.g. d'Indy).
Widor (who, incidentally, had Hungarian roots) had more influence on Vierne and Dupré, the two most central figures in the French school of the first half of the 20th century. I know Vierne was also influenced by Franck, with whom he started his studies, and one can only guess how things would have evolved if Franck had lived a little longer. But as things are, I'd say he's more of a Widorian than a Franckist.
For Tournemire, it is of course the opposite. But even so, Widor has had some influence on Tournemire (though he would have hated to admit this in later life): the strict legato (advocated in Tournemire's treatise as much as by Widor and Dupré), double pedal lines, simultaneous playing with one hand on two manuals (a device he frequently used in l'Orgue Mystique) all are things he definitely learned from Widor. Even the use of plainchant, a key ingredient in Tournemire's oeuvre, goes back to Widor rather than Franck: the Symphonie Romane, probably Widor's most inspired work, must have influenced Tournemire in some way.
Let me finish with an anecdote: it is said that when a professorship for composition became vacant in the Paris Conservatoire in 1881, Franck urged Widor to use his political connections to get the position for him, and that Widor could then take over the organ class (which Franck held since 1872). Sadly, Widor decided no to follow up on this - a regrettable decision, we may now say: can you imagine what it would have been like if Widor (who eventually succeeded Franck in 1890) had taken charge of the organ class a decade earlier, and above all if Franck had become professor for composition!