What you are seeing as gainy on the screen is actually noise from the camera sensor. Low light means low signal for for the sensor, hence you see the noise. With more light, your signal is higher but the sensor noise is still at the same level, so the graininess will go away.
I think the reason that you do not see the graininess from taking a photograph but see it in video mode is due to exposure time. I do not know 100% since I do not work for Apple, but here is my speculation based on my experience in digital photography.
When taking a photograph in a low lighting situation, the camera can take a longer exposure time (1/2 seconds lets say), which mean it can use the sensor in a less sensitive and less noisy mode, so the photo comes out clear and grain free. But when you switch to video mode, the camera can no longer take a longer exposure for the video because videos is continuously being streamed (continuous meaning at least 50-60 frames per second, likely >100 frames per second). This means the images captured by video mode must be taken at least at 1/60 of a second. In order to get a properly exposed image (an image you can see that is not too dark or too bright) the sensor is forced to run at a more sensitive thus more noisy mode. Hence the grains you see on your screen.
hope this helps :-)