Taken into consideration that diets would be the same, and both are done in equal amounts.
(eg..1 mile of swimming, 1 mile of running)
Which would burn fat faster?

Swimming burns roughly 700 + calories an hour depending on skill/intensity, whereas running is like 600. Swimming a mile in an hour places your heartbeat at about 60 - 70% of their max heart beat which is optimal for fat burning. Running places it anywhere from 75-90%, which isnt great for weightloss.
Swimming would seem to be the most effective weightloss activity based on those numbers. Its also a no impact sport, and works far more muscle groups depending on the strokes(s) you chose, therefore you would develope your physique and lung power as a whole, rather than just your legs and lungs with running.
Swimming is better if you can stick with it and perform the strokes properly over time, and you dont have to suck back tail pipe exhaust like you would if jogging along a roadway.

It depends on your skill level of each. A person with avid swimming skills will burn more calories by running, simply because the body learns to become more efficient with practice.
Also, I???ve helped a number of clients lose weight. Here are a few pointers I???ve compiled which might help:
* The key to weight loss is a very simple formula: burn more calories than you consume. It does not matter if you are eating hamburgers, brownies or carrot sticks if you eat an excessive amount of food/calories, the excess will be turned to fat within hours. The strategy is to restrict food intake, increasing exercise or both. Obviously, the latter will accelerate the process. Once again, it doesn???t not matter whether it's Weight Watchers, Atkins, South Beach Diet or whatever, the formula is the same. The "fine print" of success, however, is the ability to adhere to a specific plan over a long period of time (i.e. keep the weight off).
* Generally speaking, it does not matter if it takes you 7 minutes to run a mile or 15 minutes to walk it the calories expended will be about the same.
* One of the problems with going on a very rigid diet with a specific goal, is that it's usually a "quick fix" - the dieter attempts to gain a temporary solution to what is typically a chronic problem. Often, the dieter tries to rigidly restrict eating by, for example, skipping meals or eating salads all day. But trying not to eat is like trying not to breathe. After a while, the body and mind rebel and, like a person gasping for air, the dieter loses control and binges. As a result of the binge, the person feels that he has failed, gives up the diet altogether, eats more to make himself feel better, puts on some weight, feels even worse, gains even more weight, decides to try another restrictive diet, and begins the whole cycle all over again.
* What is often overlooked, forgotten or dismissed, is that exercise is an essential part of any weight loss plan. As long as you keep your body conditioned, with a shift of body composition more towards lean tissue instead of fat, your metabolism will increase permanently. The more muscle and lean tissue you have, the more fat you will burn ??“ all day long, even while you are resting.
* The 10-calorie rule will enable you to lose a pound or two a week while supporting your basal metabolism (internal organ function, nervous system, etc.). The rule is to allow 10 calories a day for each pound of your present body weight. As you lose weight, you can gradually adjust calories downward to keep losing at this rate. Thus, a person who starts at 220 pounds should eat 2,200 calories a day at first. If your target weight is 180, then slowly reduce your calories to 1,800 a day.
Another more aggressive approach is to institute a deficit of 500 calories a day for seven days (3,500 calories a week). Generally speaking, 3,500 calories equals one pound of body fat.
* Excess fat in the abdomen (e.g. men) is a much greater health risk than excess fat around the hips and thighs (e.g. women). The extra abdominal fat crowds the abdominal organs and the proximity to the liver means that, when metabolized, abdominal fat can raise blood cholesterol levels and lower the body's sensitivity to insulin. Obviously, this sets the stage for diabetes. Furthermore, the risk for disease rises significantly with a waist circumference of over 40 inches for men and 35 inches for women.
* Despite what all the infomercials try and sell, there is no such thing as "spot reducing" (aka: losing fat in a particular location). The fact is, muscles do not "own" the fat which surrounds them. Since all body fat is shared by all the muscles and organs, spot reducing exercises that work only the flabby areas will not help to reduce the fat in that location. However, tightening the muscles in trouble spots, through a more-complete exercise program, may improve the appearance of these areas???especially as time progresses.
* Many dieters experience a temporary plateau after about 3 weeks some may even notice a slight weight gain. This is because they have gained water weight temporarily while they are still losing body fat. The fat they are hoping to lose must be used for energy, which requires (among other elements) water to do this. Once the water leaves the cells, it enters the blood, then the kidneys and finally into the bladder for excretion. The signal that the plateau is over is frequent urination.
Andy P Certified Nutritionist
@ PreventionProof.com

it depends on the intensity and the duration of the exercise.
Swimming will burn more based on the length of time it will take you to swim 1 mile.
Realistically you should be done running a mile in less that 10 minutes.

hum...from personal experience I think running will use more calories.
Swimming will really tone your whole body though and is easier on your joints.