Emergence (noun, 1640s) originated from the Latin emergere ("rise up, bring to light, come forth"), combining ex-  ("out") and mergere ("to dip, sink"). It evolved from describing a sudden, unforeseen occurrence to its modern usage (circa 1700s) of coming into view or becoming prominent, often in scientific contexts. 

According to the Oxford English Dictionary, the noun "emergence" was first used in English in the late 1600s, with the earliest evidence dating to before 1676 in the writing of Scottish historian and bishop Henry Guthry.  The adjective emergent appeared earlier, around 1450, while emergency was used by 1631.

Evolution of meaning:   
1640s: Originally referred to an "unforeseen occurrence" or "sudden change of condition", a sense now largely obsolete for "emergence" but retained by "emergency". 
1700s: By 1704m ut was used in astronomy and optics to describe coming out of obscuration. 
1833: Adopted in geology to describe land rising from water. 
Scientific Usage: In the 1920sm C. Lloyd Morgan popularized the concept of emergent evolution, referring to new, unpredictable properties arising from complex systems. 

Emergence differs from emergency in modern usage, as the latter has come to signify a, usually dangerous, crisis needing immediate attention.  The noun "emergence" and "emergency" were sometimes used interchangeably into the 19th century. 

Just the facts:

EMERGENCE Definition & Meaning - Merriam-Webster

EMERGENCE Synonyms: 65 Similar and Opposite Words | Merriam-Webster Thesaurus

EMERGENT Definition & Meaning - Merriam-Webster

EMERGING Definition & Meaning - Merriam-Webster