Start off by making a new file, and pasting in a background that you would like to add a snowstorm to.

Make sure your foreground and background colors and black/white, and make a new layer. Choose Filter > Render > Clouds.

Add some noise to the clouds by Choosing Filter > Noise > add noise, and using the settings below.

Blur your clouds by choosing Filter > Blur >Motion Blur, and using the settings below.

You should have something similar to mine.

Next lets make it look a little more choppy, like snow. Choose Filter > Artistic > Fresco, and use the settings below.

Change you layer mode to "Screen." Create a new layer, and repeat this process. Make 3 or 4 new layers using the above method.

Open up your animation window by choosing Window > Animation.

Depending on how many layers of snow you made, create an equal amount of frames. Click on the arrow to the right of your animation window, and choose "New Frame."

In each frame, hide all of the layers accept the background layer, and one of the snow layers. In each frame, show a different snow layer. When your finished, save your image. You should have something similar to mine.