It can be helpful to think of such words not as a parent and child (i.e. "emergency" as derived from "emerge") but as siblings (i.e. "emergency" and "emerge" as two derivatives from Latin "emergo").
"Emergency" comes from the present participle of "emergo," "emergens." In Latin, the present participle signifies contemporaneous action or action-in-progress. So an "emergency" is that which is in the process of coming up--it's the grammatical inflection, not just the meaning of "come up," that adds that sense of urgency.
"Emerge," by contrast, is pretty much a straight borrowing--it keeps the basic semantics of the Latin verb more-or-less intact, making it a more neutral term that can refer to "coming up/out" physically or metaphorically.