This is a great question. The setting is a vital piece of any story. So, if the setting were completely different, it can change the story. The key word is "can," because it is possible the story would not change even with a change of setting. Two examples would be appropriate.
Let's take the story of Oedipus Rex, by Sophocles. Even if you change the setting, it will probably be a tragedy. Unwittingly to kill your father and marry your mother are tragic events. So, a modern rendition or an Eastern rendition would most likely be tragic. In this situation, the setting may not be as important.
If we take comic story of Baucis and Philemon (Ovid's Metamorphoses), who take care of the gods, and transfer it to a place like India, it would probably not be comic. Why? India is a place that is deeply religious and they would not see this as comic at all. In their culture, reverence towards the gods is an important part of life.
Based on these two example, we can say that setting is essential, but it may or may not alter the story.