The narrator in this excellent short story is clear to present events in a detached manner, but there is also a clearly tragic tone added when the veil is referred to. Consider, for example, the part where we first see the minister with his new veil:
There was but one thing remarkable in his appearance. Swathed about his forehead, and banging down over his face, so low as to be shaken by his breath, Mr. Hooper had on a black veil. On a nearer view, it seemed to consist of two fold of crape, which entirely concealed his features, except the mouth and chin, but probably did not intercept his sight, farther than to give a darkened aspect to all living and inanimate things.
Here we see a classic example of a very matter-of-fact tone that is neutral in its description - consider how the veil is described very accurately. However, this suddenly changes with the reference to the "darkened aspect" that it gives to the minister's sight. Consider other references to the veil such as the "gloomy shade" that it gives to examine this further. Such a disparity between the main tone of the story and the references to the veil emphasise its spooky and threatening nature as it stands out from the neutral tone of the story.