I don't think they can be harmed by fire, they just don't appreciate it being shoved in their faces.
As the quote itself says, they don't fear fire - they fear those who wield it. Anyone who is wielding a firebrand in their face is obviously (a) knowledgeable about Nazgul, and (b) of sufficient stature to not become a gibbering wreck in the face of the fear engendered by them, so the act of wielding fire at a Nazgul in a sign that said foe is to be taken seriously.
In Letter 210 Tolkien touches on this:
Their peril is almost entirely due to the unreasoning fear which they inspire (like ghosts). They have no great physical power against the fearless.
In addition, fire wasn't the main reason the Nazgul fled from Weathertop. Tolkien wrote on this in an essay (The Hunt for the Ring) reproduced in The Lord of the Rings: A Reader's Companion.
[The Witch-king] had been shaken by the fire of Gandalf, and began to perceive that the mission on which Sauron had sent him was one of great peril to himself both by the way, and on his return to his Master (if unsuccessful); and he had been doing ill, so far achieving nothing save rousing the power of the Wise and directing them to the Ring. But above all the timid and terrified Bearer had resisted him, had dared to strike at him with an enchanted sword made by his enemies long ago for his destruction. Narrowly it had missed him. How he had come by it — save in the Barrows of Cardolan. Then he was in some way mightier than the B[arrow]-wight; and he called on Elbereth, a name of terror to the Nazgûl. He was then in league with the High Elves of the Havens.
Escaping a wound that would have been as deadly to him as the Mordor-knife to Frodo (as was proved at the end), he withdrew and hid for a while, out of doubt and fear both of Aragorn and especially of Frodo. But fear of Sauron, and the forces of Sauron's will was the stronger.
If not for the combination of resistance, Frodo's Barrow-sword and the instinctive cry of Elbereth, the Nazgul would have persisted in their attack. The fire would not have been enough to hold them off.