Personification (and, more specifically, pathetic fallacy) sits right at the heart of this poem, establishing a relationship between the dreary weather and the speaker's dreary mood.
Everything the speaker sees as they look around at the winter landscape seems to have a personality—and none of those personalities are pleasant. The skies are "low" and "mean," or gloomy and stingy; they've only released a single snowflake, which doesn't seem to know what to do with itself, dithering about what path to take across a muddy barnyard.
The wind is even worse: all it can do, all day long, is gripe about being mistreated somehow. Its "narrow," whining voice reflects its narrow focus on its own concerns.
And that might give readers a clue about how the speaker is seeing nature, here. It's not just that the skies are low and the clouds are mean and the wind is whiny: it's that the speaker feels low, mean, and whiny as they look out at this dismal day.
The speaker makes a connection between mood and weather explicit in the poem's last lines:
Nature, like Us is sometimes caught
Without her Diadem.
Nature, this poem's personification suggests, is indeed "like Us" a lot of the time—not just because both nature and people have good "weather" and bad, but because people read the world in the light of their own moods.