Both Uses

consequence
in
Pygmalion
(Edited)

The pathos of this deplorable figure, with its innocent vanity and consequential air, touches Pickering, who has already straightened himself in the presence of Mrs. Pearce.
Act 2 *

consequential = important
This makes him a standing puzzle to the huge number of uncultivated people who have been brought up in tasteless homes by commonplace or disagreeable parents, and to whom, consequently, literature, painting, sculpture, music, and affectionate personal relations come as modes of sex if they come at all.
Act 5 *

consequently = resultantly (as a result)

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."