The streaks are due to motion of the snowflakes. You can either speed up the shutter or use a flash. I hope you don't mind my posting one where I used flash. You can see some streaking in the background, but the flash "stopped" a bunch of them.
Nikon D40, 40mm, f/8, 1.6 sec, ISO 400.
Interesting picture. I can see that the snowflakes within the range of the flash are "stopped" yet those outside of that range but lighted by the lamp show a lot of streaking indicating how they would all look without the flash.
My picture was taken in daylight and I wonder if the only snowflakes that would not streak would be those in the range of the flash which I doubt would have reached to the trees thus showing "stopped" snowflakes close to the camera and streaked flakes by the trees.