You have two things going on here, both related. The first is that the native resolution of the display you are filming is higher than the format you are capturing in. You are immediately losing 1/3 of the image information on the iPad.
This is further compounded however by the natural anti-aliasing that filming a screen at similar or lower resolution produces. Since the pixels of your camera don't line up with the pixels on the display exactly, each pixel of the camera's video file is going to have contributions from multiple distinct pixels on the display (as many as 4). While the resolution of the video may be near 2mp, it is going to get information slightly blurred together as pixels become less sharp from this blending.
This isn't an issue when you are using much higher resolution because the amount of pixels in your image that are covering multiple pixels of the display are minimal and the majority of each pixel is sharply exposed, but in the case of a video that is 1/3 smaller, as many as 9 pixels on the display may be getting blended for each pixel in the final video.
The ideal way to overcome this is either shoot in a substantially higher resolution or use a direct output from the iPad that will preserve pixel alignment. I believe there is an HDMI mirroring option and you could then capture the HDMI feed with an HDMI capture card unless they are using HDCP on the HDMI output.