Setting the trail camera to catch the badger has been a hit and miss affair. Often there are no photos but sometimes I get a fox instead. In recent weeks I have never caught the two animals on the same night.
The fox's eyes react differently to the flash from the badger's, with a much brighter eyeshine. It is caused by the tapetum lucidum, a layer of reflectors immediately behind the photoreceptors in the retina. The reflection increases the sensitivity of the animal's night vision. It is very consistently brighter in fox photos as shown below, perhaps because its eyes are larger or its tapetum lucidum is more reflective. The fox has a slit-like pupil similar to a cat's as an adaptation to hunting in low light rather than complete darkness.
Here are a couple of recent unadjusted photos of the badger looking at the camera for comparison.
It is interesting also that the fox is very often looking at the camera in the photos whereas the badger usually seems to ignore it. I suspect the fox can hear a faint noise from the camera as the flash charges. The fox relies more on its eyes and ears and the badger on its nose.