There are many arguments to this topic, and deciding which form of exercise to do is going to depend on your own goals, fitness level, health, and duration for exercise.
When exercising, the body uses different forms of fuel which change depending on the intensity, fitness level, and duration of the exercise. The body will use calories from carbohydrate store, fat stores, and sometimes protein as a last resort. Most of the time, the body is using a combination of sources: a certain percentage of fuel from one source and a certain percentage from the other which will change as the intensity or duration changes.
It is true that the walking speed will typically recruit a higher percentage of fuel from fat stores; however, it will take more time to create significant fat burn (fat calories use). If you choose to run or sprint for the same amount of time, the body chooses to use a larger percentage of calories from carbohydrate; however, so many more total calories are burned during this form of exercise that the percentage that was lower for fat calorie use is STILL going to be higher than walking. Therefore you need to plan your exercise accordingly. If you only have 30 minutes and you are in reasonable shape, with no injuries, and trying to lose fat, you should think about running and higher intensity cardiovascular exercises. If you are not conditioned, injured, or have more time for exercise walking for a longer time such as 45-60 minutes may be a better option for you. Overall, the most calorie burn for your time will come with the higher intensity exercise.
The key here is that the body is always using some form a fuel which is important for weight loss- basically calories from somewhere. When it comes to dieting the key is calories, calories, calories. The formula is simple: less input and more output (within a reasonable amount of negative calories a week for safe weight loss and the consumed calories should be from nutrient rich sources).
Keep in mind that exercise will attribute to about 30% of weight loss and diet about 70%. Depending solely on exercise for weight loss can result in over-use injuries and burnout. It is appropriate to replenish carbohydrates, protein, and electrolytes after exercise, but you don’t want to overcompensate in food consumption, thus losing the negative calorie balance if your goal is weight loss. Lastly, remember that any form/intensity of cardiovascular exercise is beneficial for your health and should be done regularly whether weight loss is a goal or not.