What is the effect of having Gurov as the central consciousness? How would the story be different if it were told from Anna's perspective?

By placing Gurov at the center of the story, Chekhov is providing a middle-aged man's view of love and life. Gurov has become rather cynical, as life has not turned out quite the way he anticipated. Seeing things through the eyes of a young woman would help explain what she sees in the older man as well as reveal what disappointments she herself has experienced.
As Chekhov sets it up, the story lets us into many of the events experienced and sensations felt by Gurov. Something about him appeals to women, even though he looks down on them. Still he can't stop being a womanizer.
Anna's youth and freshness help account for Gurov's attraction to her, but what turns that to love? Anna seems drawn to Gurov as a more urban, worldly man, but is that really what motivates her? If we knew more about her early life, her husband, her dashed hopes of wedded bliss, maybe her reasons for going to Gurov would be clearer.
This is an intriguing question that gets us thinking about point of view. Switching perspectives in literature has a solid tradition. Virginia Woolf brilliantly re-imagined Shakespeare's Twelfth Night in her Orlando, for example. The mad, first Mrs. Rochester of Jane Eyre becomes the protagonist in Jean Rhys's Wide Sargasso Sea.