My guess, though I really don't know and am just guessing, is that there are in fact dust spots on your sensor and/or lens, and the camera is set to remove them from pictures, but doesn't do the same for videos.
Like I said, totally a guess. Easiest way to check would be to pop off the lens and check the sensor and lens glass. You can use a bulb to try to blow dust off; if that fails you can buy a wet sensor cleaning kit off Amazon or other places. Cleaning the lens, of course, is easier.
