There is no difference in meaning between playwright and dramatist although, historically, some people who wrote plays preferred the term dramatist because playwright classes them with skilled laborers such as wheelwrights while dramatist sounds more prestigeous.
Shakespeare is the best-known dramatist in the world, without question. Therefore the answer must be "none".
Playwright
There have been several.
Christopher Marlowe was considered the greatest poet and dramatist of his day. He was considered the single most influential person to Shakespeare.
David Eldridge - dramatist - was born in 1973.
Terry Johnson - dramatist - was born in 1955.
David Greig - dramatist - was born in 1969.
George Fisher - dramatist - died in 1970.
George Fisher - dramatist - was born in 1909.
John Ford - dramatist - was born in 1586.
James Kenney - dramatist - died in 1849.
Michael Morton - dramatist - was born in 1864.
Michael Morton - dramatist - died in 1931.
Thomas Hales - dramatist - died in 1780.
William Cartwright - dramatist - died in 1643.
David Turner - dramatist - was born in 1927.