Thats because the "take a picture while shooting a video" function basically just saves a single frame of 1080p video as a picture... So you wind up with a picture about same aspect ratio as the iPhones 5's screen, 16x9. Its essentially just a cropped version of what the iPhone 5's camera is capable of. There are probably some other shortcomings in regards to picture quality when using this function as well (iso maybe?).
Take two pictures, one with the camera and one with the video camera, inspect each photo, you'll see the actual photo from the camera is much higher quality. The one from the video will be 1920*1080 while the actual picture will be 3264*2448, the latter being closer to the aspect ratio of the iPhone 4 as you mentioned.
In the end, you're better off taking pictures with the camera and cropping them manually.