Turn off the Fade at a Distance box in the particles then duplicate until you get the "thickness" of the snowstorm you want, then make more all over the place? That makes it look large and intense since you can get right up next to the particles without them disappearing. Though if you want them to move faster I think you'll have to edit the particle itself. I did a quick edit where I added a -3 acceleration along the Y axis, and that makes it look like it's falling faster. You can modify that acceleration to your liking if you want it to fall slower/faster or at an angle. I'll attach the file so you can look at it.