I came across two examples on YourDictionary.com and would like to know why these sentences use different articles?
You're growing into quite the young lady.
He was getting to be quite a handsome young man.
The article choice here is really random & influenced mostly by habit.
Swap the articles in the sentences and you yield the same meanings and raise no native-speaker eyebrows.
That said, the "quite the young lady" usage does have a slightly more formal, old-fashioned connotation (think Jackie O's white gloves & pillbox hats).
I'd advise to stick with "a".
What G. Ann is said is essentially right: that the two constructions are basically interchangeable. However, in some cases they may have slightly different meanings. In particular:
You are growing into quite a young lady.
This would be understood as referencing the quality of the individual's character or achievements, just like "quite the young lady."
You are quite the young lady.
will only ever be understood in the same manner as the previous sentence.
However, consider this sentence:
You are quite a young lady.
While it could be interpreted the same way, it could also mean that the subject is particularly young.
The problem seems to be an open compound noun used as a subject: that is, a set of unhyphenated words that form a noun not given by the sum of its component words. If the first word in the compound is an adjective, it produces ambiguity. This is quite a rare situation, but in such a case, using the probably will make the meaning clear.