Treadmill walking and running will feel somewhat easier physically because the “ground” is being pulled underneath your feet. And, unlike when you are outside, there is no wind resistance inside on a treadmill. Walking or running outdoors requires more energy since you are propelling your body forward with each stride. We know from research that in order to accurately simulate the same pace you walk or run outside, you need to make sure the treadmill you are working out on is set for a 1.0% incline. A 1.0% incline will make up for the lack of wind resistance. You can better simulate outdoor running by setting your treadmill at 1% incline.
Psychologically, you may have a harder time handling the monotony or potential boredom of the treadmill. I believe it is far easier to distract yourself when running outside, and I also believe it is easier to tune into your running and work on your form and stride when you are outside. My advice is that if you are training for an outdoor race, try to train outside as much as possible to get prepared for race conditions. You will also become far better at handling the varying terrain you encounter when you are racing if you have been training outside.
However, treadmills do offer advantages. The belts on all modern treadmills are padded, which makes them a good option if you are very overweight or are more injury prone (especially with knee issues), and want to decrease the impact. Additionally, there are days in central Ohio when the weather just does not cooperate, and a treadmill inside a climate controlled environment can offer a great refuge from the inclement weather, and still permit you to get your training completed.
With regard to calories burned on a treadmill versus walking or running outdoors there is little difference, especially if the treadmill is set at a 1.0% grade. Some studies have noted a slightly increased calorie burn (less than 3%) when running outside, but many others have reported no differences. Keep in mind though those treadmills that indicate the number of calories burned are generally not accurate. Many reports suggest that treadmills and other cardio machines actually overestimate calories burned by 15% to 20%.
Best wishes for your continued success with training.