Wailing Wall vs Western Wall

Wailing Wall and Western Wall both are nouns.

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

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

Wailing Wall and Western 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

Wailing Wall (noun)	Western 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