First, lets us take a few moments and layout the pros and cons to running on a treadmill verses outside. They both are great for:
improving heart health
increasing aerobic capacity
reducing stress
burning calories
Next, lets breakdown the characteristics of each mode of exercise.
TREADMILL
What is convenient about the treadmill is the variables are always consistent. It is going to be roughly the same temperature, humidity, and running surface. You are going to be able to run without interruption (what I mean by this is if you live in an urban environment like I do, and you run outside on the streets, there are going to be times when you have to stop for a traffic light or to avoid pedestrians).
The treadmill is also good for interval training, which is nice if you are wanting to increase your aerobic capacity, burn fat, and make your workout more challenging. Some units have preset programs to choose from so you do not even have to touch the buttons while you run. The downside of running on the treadmill is you are not working as hard as when you run outside because you are essentially running in place, and this equates to burning fewer calories. Also, the scenery never changes unless you have a fancy one with a TV, but do you really want to look at another screen?
OUTSIDE
Personally I do not mind using a treadmill. It is a nice option when it's January and 5 degrees outside. But, if given a choice I am running outdoors 100% of the time. Being out with mother nature is appealing to me. Breathing fresh air and feeling the sun certainly have their benefits, but they do come with their share of challenges. For starters, heat and humidity can wear you down fast, which can make you more prone to dehydration. The same can be said for running in cold weather, because your body uses more energy to keep you warm. Unless you are running on a track, the terrain constantly changes which can increase the risk of injury (i.e. sprained ankle as well as hip and knee issues). On the flip side, this can be an asset especially if you live in a hilly or mountainous environment. This is nature's way of interval training. Finally, you will most likely burn more calories since it requires more energy to propel yourself forward when you run outdoors, assuming you are running at close to the same pace as you would on a treadmill.
In conclusion, the most important message here is that you do regular cardiovascular exercise. Whether you chose to do it on a machine or outside is your choice. Be sure to hydrate and warm up before beginning, and cool down once you are finished.