Although both methods benefit the cardiovascular system, prevent muscle and bone loss, and of course burn calories, only one forces your body to continue burning calories even after you have finished your workout. At the very beginning of any workout, your body relies on its anaerobic system, which does not use oxygen for energy production. This means that you are burning more energy than your oxygen consumption provides. However, unless we push our bodies to continue to exercise at an intensity that is too high for oxygen to be present, they will quickly switch to using our aerobic system. This system relies on oxygen for energy production, which means the body cannot reach and then surpass the lactate threshold. It is only once the body is deprived of adequate oxygen that the lactate threshold is surpassed, at which point our bodies can burn extra calories both during and after a workout. Essentially, we are want to encourage our bodies to reach a state of metabolic damage so that they must burn extra calories in the repairing process.
The lactate threshold not only applies to cardiovascular workouts, but also to weight workouts. The key is reaching maximal exhaustion in whatever physical activity you put your body through - that is when we feel "the burn," therefore that is when we expend the most calories.
Avoid injury by giving your body the breaks it needs. However, when you are at the gym lifting weights, or out for a run, be sure to push yourself. Reach exhaustion often. That is when you will see the most gains. Don't ever think that you don't have time for a decent workout - the effort you put into it is what really counts.
Here are a few anaerobic workout ideas for maximal calorie-burn:
Ramp tests (choose a high yet maintainable intensity, and increase the incline on the treadmill every 3 minutes until complete burnout; can also be done on a cycle trainer)
Intervals (alternate between running (e.g. for 2 minutes) at intense speeds with walking / jogging (e.g. for 1 minute) at lower speeds
Pull-ups / chest press / curls / clean-and-jerks to complete exhaustion
Pyramid exercises (e.g. 10 medicine ball slams & 10 push-ups, 9 medicine ball slams & 9 push-ups...down to 1 of each)