The poets and the playwrights were the same people, because plays, and especially tragedies, were written in poetry. Playwrights, unlike actors, were as well respected as poets. We have actual proof of this in Francis Meres' Palladis Tamia of 1598, in which he compares the literary figures of his day to the literary figures of ancient Greece and Rome. An excerpt:

As these tragic poets flourished in Greece, Aeschylus, Euripides, Sophocles, Alexander Aetolus, Achaeus Erithriaeus, Astydamas Atheniensis, Apollodorus Tarsensis, Nicomachus Phrygius, Thespis Atticus, and Timon Apolloniates, and these among the Latins, Accius, M. Attilius, Pomponius Secundus and Seneca, so these are our best for tragedy, the Lord Buckhurst, Doctor Legge of Cambridge, Doctor Edes of Oxford, Master Edward Ferris, the author of the Mirror for Magistrates, Marlowe, Peele, Watson, Kyd, Shakespeare, Drayton, Chapman, Dekker, and Benjamin Johnson.

The best poets for comedy among the Greeks are these, Menander, Aristophanes, Eupolis Atheniensis, Alexis Terius, Nicostratus, Amipsias Atheniensis, Anaxandrides Rhodius, Aristonymus, Archippus Atheniensis and Callias Atheniensis, and among the Latins, Plautus, Terence, Naevius, Sext. Turpilius, Licinius Imbrex, and Virgilius Romanus, so the best for comedy amongst us be Edward Earl of Oxford, Doctor Gager of Oxford, Master Rowley, once a rare scholar of learned Pembroke Hall in Cambridge, Master Edwards, one of her Majesty's Chapel, eloquent and witty John Lyly, Lodge, Gascoigne, Greene, Shakespeare, Thomas Nash, Thomas Heywood, Anthony Munday, our best plotter, Chapman, Porter, Wilson, Hathway, and Henry Chettle.

So if someone tries to tell you that being a playwright was not an admirable profession or that Shakespeare began writing poetry as opposed to plays because he wanted to be recognized as a literary figure, they are totally off the mark. Shakespeare was recognized as a literary figure because he wrote plays as much as he was because he wrote poetry.