At a railway station in central Sydney, I saw a door marked EMERGENCY EGRESS ONLY. I guess that at least 99% of such doors in the English-speaking world are marked EMERGENCY EXIT ONLY.
Egress is the slightly earlier word, dating from the 1530s. Exit as a stage direction (technically, a verb) as in ‘Exit, pursued by a bear’ dates from the 1530s, but from the 1590s it was used as a noun, as in ‘All the world’s a stage, And all the men and women merely players; They have their exits and their entrances’ and (occasionally) a ‘real’ verb.
From about 1650 to about 1850, the two words were used more or less interchangeably, but then the use of exit grew and egress declined, probably corresponding to the growth in public railway travel. Then in the early 1970s, the use of exit skyrocketed, for reasons I can’t think of, but curiously declined from 2000 to 2008. Most of this was due to the use of exit as a noun; exit really only began to be used as a verb in the 20th century.
Note that the nouns and verbs exit and egress share the same form, compared to the verb enter and the noun entrance, while ingress is used only as a noun. Curiously, the use of egress and ingress has risen and fallen together over the years.
Exit is now the ‘unmarked’ choice for doors, so someone at the railway station, NSW Trains or the construction company made the deliberate choice to use egress.
(By the way, I pronounce exit as /eksət/. /egzət/ is a recognised alternative (in fact, Dictionary.com gives it first) but it sounds unbearably affected to me.)
When I was in high school, we studied Macbeth, and noticed that the stage directions sometimes said exit and sometimes exeunt. I couldn’t work out the difference, and I even remember a classmate asking me about it. It wasn’t until I studied Latin (very part-time) at university that I learned the present tense indicative of ire, go: eo, is, it, imus, itis, eunt, so exit is ‘s/he goes out’ and exeunt ‘they go out’.
There was a story which I might have categorised as apocryphal, but the interweb confirms it as true. In the mid-1800s PT Barnum ran the American Museum in New York City. Because of its popularity, the museum was crowded and there was no way of removing the customers to allow for more. So Barnum conspicuously marked a door ‘This way to the egress’. On exiting, people found themselves out on the street.