[Pitchwife]

To the good points that have been made by everybody here I'd like to add that his whole life - and therefore at least part of the mould from which his tree grew - would have been different. Only a few examples:
He would very likely not have had any childhood experience of an unspoiled countryside (at least not the way it was like at the beginning of the 20th century).
A Catholic marrying an Anglican wouldn't have been a problematic issue anymore, so he wouldn't have needed to wait for years before he could marry Edith. What would that mean for the story of Beren and Lúthien and their quest for the Silmaril, or the tale of Aragorn and Arwen?
He wouldn't have fought in World War I (might have gone to the Falklands instead - a different kind of war).
The class structures he grew up with would have been different, so the relationship between Frodo and Sam would be different as well.
In short, he would have grown up in a different world, experienced different influences, matured into a different man and written a different book. Still a great book, maybe, but not the one we know and love.