One can only assume why Tennessee Williams chose to publish his essay A Streetcar Named Success prior to the opening of his play, A Streetcar Named Desire.
Given that the message of the essay focused upon the artist's role in society, he may have used the text to justify the actions which would be seen in the play. While this is simply an assumption, based upon the fact that Williams is not able to answer the question himself, one could easily justify that the essay was needed to support an author's/artist's role so that society could more readily accept and understand the purpose of artistic expression.
In his essay, Williams states that he had experiences with "disenchantment." What this does is show how disenchantment had impacted Williams and foreshadows the disenchantment which will be seen in his play A Streetcar Named Desire.
Later in the essay, Williams states that "one does not escape that easily from the seductions of an effete way of life." What he means is that it is not easy to escape from a life where one was pampered. Again, he can be assumed to be referencing Blanche from the play. Blanche has lived an effete way of life. Her inability to accept that her life is a mirage forces her downfall.
In the end, one can only make parallels between the essay and the play. By making these parallels, one can try to justify why Williams choose to publish his essay prior to the performance of his play.