There are several reasons, though not all of them would apply for one's usual holiday trip.
Obviously, during night its dark and this gives you a fair bit of cover if you want to remain unseen. This reason for traveling at night is common for many hunters in the animal kingdom (and sometimes also their prey), has been used by humans in insecure or hostile territory for centuries and remains a standard technique for some military operations even today.
Evading the elements
Hiking throughout the night and resting in the day can be especially useful in hot, arid conditions. For example in deserts, extreme heat during the day and severe cold in the night often come together. Hiking during the night here serves the double purpose of keeping you active and warm when it is cold while allowing you to save water resources during the day.
When conditions get worse (cold, moisture), getting up during the night and starting to hike will not only make you feel warm again, it will also boost your morale. Being active, and actively coping with a bad situation rather than staying miserable in the tent for the rest of the night will make you feel better.
You want to avoid tourists
No, seriously, this is a real case in point. Most people travel during the day and one of the reasons we (or at least, I) get outdoors is to enjoy some quietness in the wild. Depending where you are, hiking at night is your best bet to fulfill this dream.
Navigation (only very theoretical)
Consider a situation where it is all cloudy during the day (so you do not see the sun), you lack a compass and currently have no sense of direction: if you are capable of navigating with the stars, you might consider waiting until dark and hope that the sky will clear up until then.