While it may be true that the 5S has lower quality low-light video recording than other iPhones, there is a fundamental physics fact that you have to keep in mind: To get more light in a picture –controlling for other factors such as ISO and aperture– you need more time for each picture, more exposure.
I just did a test on my iPhone 6, and the "problem" persists.
The thing is, it is impossible for an image to look brighter with less exposure.
In the case of my pictures, the first picture is taken at 1/15 of a second, with an ISO of 500.
I can't find the ISO information on the video but I know the exposure: it has to be at the most 1/60 because I've setup video to have 60 frames per second. So it is 4 times faster -or 4 times less exposure- which explains the darkness.
Remember a video is just a series of pictures.
From the comments one of the links provided by Doc G.(emphasis mine):
When taking a digital image, there are 3 main factors - aperture (or f-stop), shutter speed and ISO (sensitivity). In manual mode on a 'proper' camera you can experiment with all of these to achieve a desired result. In automatic mode, the camera makes all the decisions hoping to manage a decent (fast) shutter speed to combat camera shake or motion blur, a low value ISO to reduce noise, and an appropriate aperture to get the best exposure.
Unfortunately, when shooting video one of these factors is set in stone. At 30fps your shutter speed will be set to 1/30th of a second, and at 120fps this will be 1/120th of a second. If you are talking about low light conditions, then you are required to open the aperture or increase the ISO to compensate. The iPhone 5s has an aperture of f/2.2, so that becomes another limiting factor. The only place left to go is an increase in ISO - meaning more noise.
Therefore the iPhone 5s is either hitting it's (reported) ISO 2500 maximum, or deciding on a compromise value to avoid excessive noise. In other words, if there isn't enough light to be shooting video, you shouldn't be trying to shoot video. At least not until sensor technology improves, which will happen eventually.