Both Uses of

consequence

in

Medea, by Euripides - (translated by: T.A. Buckley)

not, by which thou art grated, loathing thy bed, nor smitten with desire of a new bride, nor having emulation of a numerous offspring, for those born to me are sufficient, nor do I find fault with that; but that (which is of the greatest consequence) we might live honorably, and might not be in want, knowing well that every friend flies out of the way of a poor man; and that I might bring up my children worthy of my house, and that having begotten brothers to those children sprung from thee, I might place them on the same footing, and having united the family, I might flourish; for both tho†

consequence = importance

But my children, enter you the wealthy palace, to the new bride of your father, and my mistress, entreat her, beseech her, that you may not leave the land, presenting these ornaments; but this is of the greatest consequence, that, she receive these gifts in her own hand.†
*

Definitions:

(1) (consequence as in: a direct consequence of)
a result of something (often an undesired side effect)
(2) (consequence as in: of little consequence)
importance or relevance
(3) (meaning too rare to warrant focus)
In classic literature, consequential may refer to someone with too much feeling of self-importance as when Dickens wrote "Because he's a proud, haughty, consequential, turned-up-nosed peacock."

Self-consequence was used in a similar manner, but is more easily understood by modern readers since important is one of the modern senses of consequence.

Another classic sense of consequent that is similar to importance or significance refers to "material wealth or prominence" as when Jane Austen wrote: "They had each had money, but their marriages had made a material difference in their degree of consequence."