What is the difference between courting a woman and 'dating' a woman (or, rather, a girl who may be chronologically adult)?
Courting is *explicitly* aimed at marriage. When a man is courting a woman, he is *explicitly* announcing that he thinks she is a suitable marriage partner; the fact of the courtship is to get to know one another well enough to decide whether to make the mutual commitment.
On the other hand, 'dating' isn't aimed at anything. Well, at its (limited) best it isn't aimed at anything (*); usually it is aimed at getting her (**) in the sack for no-committment sex.
(*) "Let's see where this goes" is not a plan, and it rarely leads to success. Men and women being men and women, "where this goes" generally turns out to be the back seat of a car, or some similarly ridiculous place to "make love". And that generally leads to getting married to an unsuitable person, which then generally leads to divorce.
(**) Edit: I don't mean to imply that women don't also try to use men for less than noble ends, including using them as walking dildos.