I shot a video using both an ND and a polarizing filter (not at the same time).
The ND filter was used to reduce the light so I could employ reasonable apertures, not overexpose and keep the shutter at the optimal (for 108060p shooting) 1/60th. But the filter also protects the lens. From what, you might say? Well, in this case from flying ice. Here is what would have happened to my lens (this is a framegrab from the video below, where you will see what happened (sharp stop on ice)):
I just wiped the ice and water off the filter and I continued. A UV filter would of course offer the same protection, but does not do much for video quality.
The polarizing filter, among other things, enables you to reduce glare when shooting through windows. In this case I was shooting the Christmas display windows on Fifth Avenue, a popular site. On a bright day, the windows are just a mirror of glare. With adjustment to the filter, you can see through the glare (there are still reflections). Here is a frame grab example of a shot through the window (everything you see is behind glass):
And here is the video, with the near-catastrophe in HD and full color:
You see a Christmas tree, buildings, ice skating, and those Fifth Avenue holiday windows. Almost all accompanied by holiday music, through speakers. There is Christmas music everywhere - no soundtrack needed. In the last shot, in the deep, dark subway I had forgotten I still had the polarizing filter on (which reduces the light), so it is grainier than usual but still fine. Music there too, but live.