How are Sylvia and the stranger similar? How are they different? Why did Sylvia not tell the stranger about the heron?

Sylvia and the stranger are similar in that they both have (or claim to have) a real appreciation for the beauty of nature.  Just as Sylvia has a special appreciation for the birds, so does he.  He tells her grandmother, "'I am making a collection of birds myself.  I have been at it ever since I was a boy.'"  Further, he says that he cannot think of anything he'd like better than to find the white heron.

However, Sylvia and the stranger differ greatly in how they express their appreciation of nature.  For Sylvy, who is often compared to a flower, a move to the country from the city was a permanent once because

Everybody said that it was a good change for a little maid who had tried to grow for eight years in a crowded manufacturing town, but, as for Sylvia herself, it seemed as if she never had been alive at all before she came to live at the farm.

Living in the country has enabled Sylvia to grow and thrive when she could not in the city.  For the hunter, on the other hand, his time in the country is only a "vacation" that he uses to hunt.  While Sylvia's heart "beat fast with pleasure" when she heard the birds singing in the bushes, the hunter intends to "stuff and preserve" them.  The both appreciate nature, to be sure, but the child expresses her appreciation by observing these animals, alive, in nature and the stranger expresses his by shooting them and taking them back to the city.

In the end, Sylvia does not tell the stranger about the heron because

The murmur of the pine's green branches is in her ears, she remembers how the white heron came flying through the golden air and how they watched the sea and the morning together, and Sylvia cannot speak; she cannot tell the heron's secret and give its life away.

Although the hunter offered her $10 to help him to find the bird -- an awesome sum to her mind -- on some level, this child recognizes that the value of the "pine's green branches" and the "golden air" and the beating heart of the heron who shared her beautiful sunrise are of far greater value than money.