The linked question on biology.SE actually is for many aspect already the answer, one just needs to convert the findings to the outdoors/hiking. I will first reiterate the most important anatomical points from the said question and then pull conclusions for outdoor use of red light. So if you don't care about the background and know what rods/cones are, just skip the next paragraph or if you want more, go read the linked question instead.
The eye contains (primarily) two types of photoreceptor cells: Rods and cones. There are three types of cones sensitive to different wavelength areas, thus allow for color vision. They are active during good illumination (photopic vision). Rods are much more light sensitive but there is just one type, so distinction of color is not possible (scotopic vision). Rods are primarily sensitive in the blue-green light range. In intermediate lighting conditions both kinds of cells can be active (mesopic vision).
The distribution of rods and the different cones in the eye is not at all uniform: Cones are concentrated in the center while rods are absent in the center and their maximal concentration is at a visual angle of approx 20deg. Additionally in the center there are no S-type cones (sensitive primarily in blue light) and more L-type (redish) than M-type cones
A great (while lengthy) source for information about the sensitive of the eye to light at different wavelengths is Color Vision by Bruce McEvoy.
Red light affects rods and thus night vision the least. It still does, so light intensity is key. What is more important is that the center of the eye is mostly sensitive to red light and has no rods at all. So it is impossible to read anything with blue light or in very dark conditions where only rods are active (well the second is anyway problematic for reading). Just try to read without moving along with your eyes.
Using dim red light allows to use rod cells for peripheral low light vision and keeps them active (dim red light!), and at the same time you get much increased vision in the center. Thats why red is great for reading maps in the dark.
That one is quite obvious: If it is important to discriminate colors you need to use white light.
Red light is might be useful for stealth as many animals have more rods than cones compared to humans, and are thus perceive red light relatively less strongly compared to humans. This might give you an advantage.
Visibility at distance
When light travels through air it is scattered by a process that is dependent on the wavelength. Short wavelengths (i.e. blueish) are scattered much more than red (that is why the sky is blue and sunsets red). Thus red light is better visible at a distance. Uses are position lights, e.g. car rear lamps or the light on the back of the Petzl Nao+ headlamp.
Also there seems to be a misconception about the brightness and cone characteristics of a light source: These are independent of the color. You could have a red light just as bright and focused as a white one. It is clear that a bright red light is not useful, as it would destroy night vision as well and reduce color perception, while I cannot think of an advantage. However having a more focused red light would actually be preferable to the wide cone usually used for head lamps. As described above, red light is most effective for the center of the eye, so only illuminating a spot in direction of view would be more effective.