Of course. We see and hear the entire story of the play through the lens of Tom, our narrator. In addition to the points made above, the most important difference for me would be my view of Tom. Imagine not knowing what we know about what he's thinking or what has happened outside the course of the play. He would look like a shirker and a derelict in ways that probably aren't fair to label him with. True, he's got little motivation and his behaviors aren't always commendable, but with him as narrator at least we understand why.
I agree that I as the reader would view the play differently if there were no narrator. Adding a narrator to the play immediately begs the reader to consider the narrator's reliability. Tom is recalling the events in the play, so the reader must question how selective Tom's memory is and whether or not the reader is receiving all the events in the exact manner in which they occurred. There is always some level of doubt in considering these questions, and this doubt causes the reader to appraise events differently.
Importance of Narrator
Would your appraisal of the events be different if there were no narrator?
By using Tom as the narrator, Tennessee Williams tells his play from the retrospective point of view. Tom is really two characters in the play: the young Tom enduring life in that apartment and the older Tom who is remembering that past. As the narrator, Tom understands much more about himself and about life than he was able to understand before leaving home. There is sadness in him where before there had been anger and resentment and longing. In his closing speech as the narrator, Tom brings a special poignancy to the drama as even Amanda is shown with "dignity and tragic beauty." Without Tom's retrospective point of view, many of Williams' themes and greatest ironies could not have been realized.
Some other modern American writers who also used the retrospective point of view very effectively are Scott Fitzgerald in The Great Gatsby, Harper Lee in To Kill a Mockingbird, and Edith Wharton in Ethan Frome.
Absolutely. Since it is a play of recollection, not having a narrator would cause the play to lose it's personal touch. Because there is so much weighing on Tom to support his family while his mother concerns herself with other things (Laura's suitor),his perspective is crucial in understanding the differing priorities in this family.
There simply would be no play without Tom, the narrator, because, not only is he the narrator, he is also a character in the play, and supposedly the author of the it. As narrator, Tom can look both backward and forward and discuss the ramifications of events as they take place. As a character in the play, the audience can sympathize with his plight and see the family dynamics in action. As the supposed writer of the play, we have to realize that the play is told from Tom's memory so, in some cases, that memory may be hazy or unreliable. That leaves it up to the audience to watch the action carefully and to decipher the play's themes.