Unfortunately, there's no simple and easy answer to this question. On one hand, I want you to get as many minutes of exercise as possible during the day, and if that means two 30-minute sessions instead of one 30- to 45-minute session, then two sessions is a better choice. On the other hand, you can make bigger physical adaptations, which lead to greater weight loss and greater improvements in endurance, strength, power, and cardiovascular health, by exercising for one sustained 60-minute session instead of two 30-minute sessions. Which brings us back to your original questionwhich is better for you?
If you're going to be exercising for 60 minutes anyway, and the only difference is whether it's in two sessions or one, I'd prefer to see you do one 60-minute session. Sustained exercise for longer periods of time leads to important changes in your muscles. The mitochondria, which are the powerhouses that burn carbohydrate and fat during aerobic exercise, grow and multiply in response to longer and harder bouts of exercise. If the mitochondria you have are already capable of delivering the energy you need for 30 minutes, then there's no reason for your muscles to adapt and develop the ability to burn more fat calories per minute.
Remember though, exercise is about more than burning caloriesit's also about challenging yourself to the point where your body has to adapt and get stronger. Longer exercise sessions can be an important part of this equation. If you don't have time to complete a 60-minute session and have to split your exercise into two 30-minute sessions, increase the pace/intensity of your shorter workouts so you're challenging your body.
And as for burning fat, there might be a few minutes at the very beginning of exercise where you're not burning that much fat, but that's only because fat has to be burned by the aerobic system and this system takes a little while to break fat down into usable energy.
In the meantime, you're burning more carbohydrate because your body can break it down to energy faster. However, after a good warm-up, you're aerobic system is fully up to speed and able to burn fat at its maximum rate. The more developed your aerobic system is (more and bigger mitochondria I talked about previously), the quicker your body will start to burn fat.
So, the biggest lesson is to focus on sustained, challenging exercise to improve the strength of your aerobic engine. With a well-tuned engine, you can burn a lot of fat every time you exercise. Focusing on burning fat before you focus on developing a good engine is putting the cart before the horse.