As several astronomers and casual observers pointed out in the replies to landru79's original tweet, the "snowstorm" depicted almost certainly isn't a true snowfall of the sort experienced on Earth and other planets. Instead, there are likely two or three different phenomena creating the snowy effect.
Up close to the camera, dust particles backlit by the sun are likely moving around, mimicking the look of snow on Earth. Cosmic rays may also be creating snow-like artifacts on the images. And those dots in the background, that appear to be falling straight down and disappearing behind the cliff? Those appear to be stars, which look like they're falling because the comet is rotating as it orbits the sun every 6.5 years.
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