Western Wall vs Wailing Wall

Western Wall and Wailing Wall both are nouns.

Word	Noun	Adjective	Verb	Adverb
Western Wall	Yes	No	No	No
Wailing Wall	Yes	No	No	No

As nouns, Western Wall and Wailing Wall are synonyms defined as:

Western Wall and Wailing Wall: a wall in Jerusalem; sacred to Jews as a place of prayer and lamentation; its stones are believed to have formed part of the Temple of Solomon

Western Wall (noun)	Wailing Wall (noun)
a wall in Jerusalem; sacred to Jews as a place of prayer and lamentation; its stones are believed to have formed part of the Temple of Solomon	a wall in Jerusalem; sacred to Jews as a place of prayer and lamentation; its stones are believed to have formed part of the Temple of Solomon