Your friend is right Liz. That's caused by the flash highlighting the snow.
What you're trying to do is not easy. The best thing would be to suppress the flash and take the shot with the camera supported on something (preferably a tripod). The only thing with that is that the shutter speed would then necessarily be so slow that the snow would blur and show as faint streaks rather than "freezing" the motion.
Here is what I would try: First, I would try it on a tripod. If you have much control over your camera, I would experiment with differing shutter speeds. You may find that the nearest snow flakes would blur so much that they become invisible.
The other thing I would try would be to try to cover the first few feet above the camera (take the photo from undercover somewhere) and still use the flash. In this way, there would be no near flakes to catch the light.
Either way, the flash needs to be used either not at all or only to "catch the highlights" (Slow shutter sync). Outside night shots always mean no flash unless you're trying to illuminate close objects (snapshots of people).
Basically, if your camera has much manual control - experiment
I hope this helps some - let us know the outcome.