Looking purely from a calorie expenditure point of view, it is far more beneficial to complete a distance in the shortest amount of time.  What happens is that, by pushing yourself to work at a higher output, you will burn a higher degree of calories in a shorter period of time.  You will also fatigue your body considerably more and thus your body will require far more recovery for many hours after your workout.  During this time, you will be breaking down fat tissue at a higher rate.

From a fitness perspective, you improve your level of fitness by exercising outside of your level of comfort.  Walking for a particular distance will have minimal fitness benefit, but pushing yourself by running will boost your cardiovascular fitness significantly.

While both the fat loss and fitness benefits are a result of the increased degree of recovery after your workout from the higher level of intensity, if you are feeling fatigued, this would be an important reason to back off the intensity.  In essence, we need to find a balance.  Too intense and you will run yourself down (excuse the pun).  Not intense enough and you will not obtain the desired weight loss results.

One other consideration.  Running will induce far more impact throughout the joints (ankles, knees, hips in particular).  If you experience any discomfort throughout these joints, then you would want to avoid the high impact nature of jogging.  This is often very important for people who are overweight as, the heavier you are, the more joint impact you can expect to experience.