Let's keep it simple. Losing weight is simply calories in versus calories out. That being said obviously that fastest way to lose weight is by cutting your carbohydrate intake and increasing your net output value in cardio/circuit training/training!
Many things need to be considered when you try to lose weight. The body is typically a habit machine which is why some crash dieting goes horribly wrong for many individuals. You play around with your insulin spikes and sure enough you gorge one day after a week of dieting and gain the weight right back. Doing this correctly requires discipline by simply slowly cutting back you carbohydrate intake and slowly increasing your aerobic capacity bit by bit. Simply running 10 miles the first day after a delayed onset of working out will result in a heightened result of over-training and the body will be put through a serious shock that will result in unneeded weight gain and injuries.
The extent to which you burn calories depends on the intensity of your workout. A 180pd athlete squatting 315pds for 5 with a max of 355 is hitting a high percentage of his maximum and thus is rapidly burning calories. Going on a 45 minute "light" jog will only give you so 'fat-burning'; all depends on your intensity. Hence you see coaches prescribe fartleks or speed endurance workouts to athletes who need to lose weight (on top of diet, sleep, increasing work capacity, etc. etc.; it's a combination of a lot of things).
Overall you need a balanced lifestyle. Sitting 15 hours staring at the computer results in a sedentary lifestyle which many regular people are doing day in and day out. Consistency, discipline and self motivation with a basic healthy structure will result in optimal performance. Just like your mommy told you...eat your greens. The basic principles are still the same. Our society is just caught up on overconsumption and trying to look for the secret answer to losing weight. "Cardio is the one answer" or "It's all about diet" is complete and utter nonsense. It's an accumulation of everything in your lifestyle as well (stress from work can also alter your weight loss and gain). Balance it out, don't overconsume, eat a variety of good food, stay active. That's all you need to know. When you figured that out then ask what will give you an extra edge to be the top 0.000001% of athletes in the world.