Running on a treadmill feels somewhat easier physically because the ground is being pulled underneath your feet and there's no wind resistance. On a treadmill, the smaller stabilizer muscles in your lower legs don't have to work as hard.
Running outside demands more from your body because you're propelling your body forward stride for stride. Your calf muscles have to work much harder than when you're on a treadmill. As a result, many runners find that they experience calf muscle pain, shin splints, Achilles tendonitis, and other issues when they run outside after exclusively running on a treadmill for a long period of time.
Treadmills are a good option when the conditions are not conducive to running outside- too hot, rainy, etc. Many treadmills are padded, making them a good option if you're overweight or are injury-prone (especially with knee issues) and want to decrease the impact. You can also easily track your distance and pace, and set up your pace and course to be exactly what you need.
If you're training for a specific outdoor race, try to train outside as much as possible to get prepared both physically and mentally prepared for race conditions. Your body needs time to adapt to the stresses of running on pavement. The only way to make sure that you're ready for an outside race is to run outdoors.