Nazgûl is Black Speech and is translated as Ringwraiths or sometimes Ring-wraiths. Nazg means ring and gûl means wraith/spirit in the broadest of terms.
gûl is a loan from the "Black Speech" and refers to evil and necromantic arts. Cf. Nazgûl (nazg-gûl).
Parma Eldalamberon XVII, "Words, Phrases and Passages in Various Tongues in The Lord of the Rings", pp 31
nazgûl. cf. nazg = ring. gûl, (phantom, shadow of dark magic, necromancer), slace, servant? nazggûl.
Ringwraiths is a translation of Nazg-(g)ûl < nazg, Ring in the Black Speech, and gûl, an [evil] spirit under the control of Sauron.
Parma Eldalamberon XVII, "Words, Phrases and Passages in Various Tongues in The Lord of the Rings", pp 79
(Gûl in Black speech for a 'wraith' is probably derived from the Sindarin: as nag-gûl, nazgûl, Ringwraith.)
Parma Eldalamberon XVII, "Words, Phrases and Passages in Various Tongues in The Lord of the Rings", pp 125
If that wasn’t convincing enough this quote also explicitly confirms that both are the same.
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 downfall. 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"