|MadSci Network: Biophysics|
The short answer to your question is that no, running and walking don't burn the same number of calories. The longer question is why not and how much. Both of these are very interesting questions that have been studied in much detail. In order to get a better understanding of the answer to your question, I'll briefly touch on these.
First, the amount of calories any exercise burns depends on the body composition of the individual who is doing the exercising. Different ratios of lean tissue (i.e., fat-free tissue) to body fat lead to different amounts of calories being burned. The more lean muscle tissue you have, the more calories you'll burn. The more body fat you have, the less calories you'll burn. And this isn't just true for "exercise calories"; this applies just as well to your Resting Metabolic Rate (RMR). Also, you burn more calories when you exercise using the big muscles, like the large leg muscles during walking and running, as compared to the smaller stomach muscles during, say, situps (btw, this is one of the reasons why running burns more calories than walking: it calls more muscles into play). The amount of calories burned by a average man while walking (approx. 90 calories for a mile) or running (approx. 100 calories for a mile) are pretty close (since running a fixed a distance is much faster than walking that same distance) but the difference becomes more significant if you do both activities for the same amount of time. For example, the amount of calories burned for either exercise, when it's sustained for an hour, can be found here.
Okay, so that offers a little overview of how many calories each exercise burns. Now to the why. There are really two kinds of fuels that the body burns: one is sugar (in the form of stored glucose, or glycogen) and the other is fat. Exercises that are of a longer duration burn more fat whereas short-term expenditures of energy are mostly fulfilled by the glucose stores. The exact breakdown of this is as follows: for about the first minute of an exercise, the body uses adenosine triphospahte (ATP) and creatine phosphate (CP) directly. If the exercise lasts up to 5 minutes, the body starts calling on the reserves of glycogen and burns them via anaerobic glycolysis. Finally, if the exercise lasts beyond about 10 minutes, the body uses a mix of anaerobic (glycolysis) and aerobic (fat) metabolism. And then if an exercise lasts an even longer time and involves the big muscles (e.g., a marathon!), the body uses stored fat as the primary energy provider. This is , of course, a gross simplification of the actual biophysical/biochemical processes; but you can still think of the ultimate fuel in all this as ATP since glucose and fat are burned to synthesize ATP. Additionally, exercise increases the RMR thus burning even more calories.
On a side note, a kilocalorie is commonly called a calorie; so when a box of cereal tells you that it has 100 calories per bowl, it really means that it yields 100 kilocalories. Now the different fuels provide provide different amounts of energy upon burning. ATP provides 7.3 kcal/mole and since glycolysis yields approximately 36 ATP per molecule of glucose, that gives approx. 263 kcal from glycolysis of one mole of glucose (btw, a good reference for this is here). Put another way, for each gram of glucose, you get 4kcal. But for each gram of fat, you can get 9kcal. To see why, please check out this site. So, to wrap this up, the amount of calories you really burn also depends on what kind of exercise you do (i.e., which muscle groups are called into play), how long you do it (i.e., duration and intensity), and what kind of reserves are being recruited (i.e., body composition).
Well, I hope that helped (although that might have been a little more information than you wanted). If I forgot to properly define any term, don't panic; you can find a online glossary here. Also, if you'd like to search for more information on any of this, an excellent starting point is Yahoo, which is where I found the above links. Good luck in your search and please feel free to drop me a line if you'd like something clarified further!
Try the links in the MadSci Library for more information on Biophysics.