This simple choice, switching the point of view of the narrator, has striking effects. Checkov's story seems more romantic, reflecting the older male character's desire for youthful love, whereas Oates' story seems more desperate as the female character attempts to escape a bad relationship and to try and manage a personal crisis.
As an initial matter, the narrative techniques are quite similar. The narrators are external to the stories themselves, removed from the action and detached, and limited in knowledge in each story. Both authors rely upon the third person limited narrative technique. The narrative technique is limited rather than omniscient because each author tells the story from the point of view of a single character. In Chekhov's story, the reader learns more about Anna through the eyes and thoughts of Dmitry; in Oates' story, by contrast, the reader learns more about Dmitry through the eyes and thoughts of Anna. In short, the basic narrative framework, third person limited, is similar in both stories.
The specific point of view employed by the authors, though, results in very different interpretations. Chekhov's protagonist, Dmitry, is aging, unhappy in his own marriage, and an adulterer of some stature. He possesses a low opinion of women in general.