Curiosity does have a video camera. It can take HD video at 10 frames per second.
The video is 1280 by 720 pixels at a rate of about 10 frames per second. If set to individual frames, then they are 1200 by 1200 pixels. The CCD is 1600 by 1200 pixels. Details here: MSL Mast Camera (Mastcam) Description
An obvious thing to video is a dust devil. This is a video that is made by combining a series of images taken by Spirit, spans 9 minutes and 35 seconds on Spirit's sol 486.
And this is a video Curiosity took of Phobos crossing in front of Deimos - but at around one frame per second time lapse:
Does anyone know if it has ever used this capability? And why do they do time lapse sequences when they could do an HD video?