No. Tolkien does not explain the difference but likely the difference in power between the One Ring and the Nine Rings as well as their original purpose defined this difference in ability.
The Nine Rings of Men did exactly what they promised, they gave power to their bearers. Power they were eager to accept in order to feel equal to the might of the Elves. Since we are given very little information about the Nazgûl and their time as living men, we are forced to make assumptions about the rings and their abilities.
The Men who gained these rings were likely powerful, ambitious, arrogant and incautious. The Men of Númenor certainly were.
We only know the men were both kings and sorcerers, which implied they had some mastery of magical ability. Given Tolkien's world, what that looks like might vary, but we can assume the nature of the Rings increased those leadership and magical abilities (as humble as they might be). But knowing how difficult magic appeared in general, ANY magic might be enough to turn the outcome of any single battle.
Later in the Second Age, Sauron gave the Nine to powerful men, kings and sorcerers, including three from Númenor, all of whom fell swiftly under the rings' domination. They became the Nazgûl or Ringwraiths: spirits of terror whom Sauron could command even without the One. Their lives were extended indefinitely by the rings, and they became Sauron's chief servants, especially during the first part of the Third Age when he was too weak to act on his own behalf.
Any of these Rings of Power seemed to render a man who wore it invisible. The Nazgûl could not be seen directly by mortal eyes, but wore dark cloaks to give themselves form. Frodo saw their true form when he put on the One Ring.
So the rings were given as gifts to men. It is safe to assume they did not give men invisibility all the time, lest they would never wear them except under duress. For Sauron's trap to work, they had to be wearing them.
Likely all of the Men who were given the Nine fell to their power at the same time, because their transformation might have given rise to one of their number escaping the trap. As an alternative if they died in battle or intrigue, their spirits could have been trapped and reborn as Nazgûl increasing the terror their legend might cause.
Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had, as it seemed, unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men; but too often they beheld only the phantoms and delusions of Sauron. And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore and of the domination of the One which was Sauron's. And they became forever invisible save to him that wore the Ruling Ring, and they entered into the realm of shadows. The Nazgûl were they, the Ringwraiths, the Úlairi, the Enemy's most terrible servants; darkness went with them, and they cried with the voices of death.
— The Silmarillion, "Of the Rings of Power and the Third Age", 346
I suspect the Nazgûl after wearing their rings for a while became permanently invisible, moving into the realm of shadow a bit at a time. They probably transitioned after they exceeded their normal lifespan becoming shadows of themselves, i.e. 'spirits of terror' previously described.
So their voluminous cloaks may not have been anything other than a place holder for the living to be able to hold a conversation with this otherwise disembodied spirit whose power now was the ability to affect the world of the living with either their limited magic, fear-causing ability or their seeming invulnerability to mortal weapons.
If a Nazgûl wanted to be invisible to mortal eyes, all he had to do was to drop his cloak. Only the sensitive ears of an elf, or the bearer of the One Ring would have any chance of detecting them since they no longer truly existed fully in the world of the living. This state probably wouldn't need the actual ring once they died, since their spirits would be bound by having worn the ring and were now tied to Sauron anyway. This is why the waterfall was merely an inconvenience during the Rivendell incident.
