The term is not a variant spelling of the common misspelling "playwrite": the word wright is an archaic English term for a craftsman or builder (as in a wheelwright or cartwright). Hence the prefix and the suffix combine to indicate someone who has "wrought" words, themes, and other elements into a dramatic form - someone who crafts plays. The homophone with "write" is entirely coincidental.
The first written record of the term "playwright" is from 1605, 73 years before the first written record of the term "dramatist". It appears to have been first used in a pejorative sense by Ben Jonson to suggest a mere tradesman fashioning works for the theatre. Jonson described himself as a poet, not a playwright, since plays during that time were written in meter and so were regarded as the province of poets. This view was held as late as the early 19th century. The term "playwright" later again lost this negative connotation.