Since the question has come up in Comments, let's clear up this playwright/playwriting confusion once and for all.
The term playwright bears this wonderful anachronistic notion of the dramatist as "handicraftsman." (Hence "wright" as in shipwright.) It's very much in the same vein as dramaturgy, which, as Michael Feingold once helpfully pointed out in a column, is an etymological cousin of metalurgy.
According to the Oxford English Dictionary, "Playwright" dates from 1616--yes, the very year of Shakespeare's death. But it was his rival Ben Jonson who is credited with coining the term (or at least authoring its first extant appearance):
I'll leave it to the more adventurous antiquarians out there to unpack that.
1616B. JONSON Epigr. xlix, in Wks. I. 781 Play-wright me reades, and still my verses damnes, He sayes, I want the tongue of Epigrammes; I haue no salt: no bawdrie he doth meane. For wittie, in his language, is obscene. Play-wright, I loath to haue thy manners knowne In my chast booke: professe them in thine owne.
Note the hyphenation. It also appears as such in OED's next historical example...
...where apparently this guy Clifford is telling Dryden to avoid the profession, even back then!
a1677M. CLIFFORD Notes Dryden's Poems (1687) iv. 16 Wherein you may..thrive better, than at this damn'd Trade of a Play-wright
My off-the-cuff historical diagnosis is that we see here the growth of the theatre business in 17th century England, after such hitmakers as Shakespeare and Jonson proved one could make a profession out of poetry for the stage. Concurrently in this period you also will see dramatists referred to flat out as "poets," in Restoration prologues, for instance. ("Our poets," etc) But that fell by the wayside once plays turned pretty consistently to prose by 1800. And so "playwright" was all that was left, I guess. And so it remains.
Notice also how the hardwiring into the word of menial craftsmanship--as opposed to ethereal poetry and high art--coincides with that ol' anti-theatrical prejudice. While many a dramatist to this day would be proud to be deemed a good craftsman, they don't share that honor with would-be "novelwrights" and "symphonywrights." Come to think of it, "Dramatic Composer" doesn't sound all that bad, does it?
As for the verb form, "Playwriting" is totally legit since it's actually a different word, not "playwright" modified--that is, "writing plays" as opposed to being a "wright" of plays. OED does recognize "playwrighting" but only dates it from 1892--which is puzzling since "wright" as a verb seems to have gone out of style around...oh, 1616. "Playwriting" actually appears to predate "Playwrighting" which shows it's not some corruption. Maybe putting that hyphen back in would help: play-writing?
We also might be interested in reviving what Carlyle in 1831 termed the "playwrightess."