Running outside will help you not
fall asleep. Whenever I run on a treadmill, I feel it am the one keeping the electricity on in the gym. Also, it always seems like the times and distances on a treadmill are screwed up. For instance, I can run two miles in about 13:30-14:00, but on a treadmill it took me almost 17:00 - what a crock.
As for weight
losing weight, I don't think your body can tell the difference as long as your heart rate is elevated. endurance wise, you'll be able to run farther on a treadmill. I have heard ppl say that after running on a treadmill, they still aren't good at running outside, so I would say its pretty specific.
FWIW - running outside will prevent you from cheating - when you are 2 miles away and the only way to get back is to run, what are you gonna do? As opposed to a treadmill, you can just get off and go back to the parking lot IMO