Extreme temperature hot or cold will yield a greater caloric output, because your body has to work harder to keep your core temp regulated, however the difference is negligible. The title of your thread is a bit misleading because exercising is broad, and running is specific. Running outdoors is more difficult because of uneven terrain, however if you're on track in both scenarios, again it doesn't really matter.
Is there a benefit to sweating? I guess there's that whole "sweating out the toxins" thing, but it just means you'll have to work harder to stay hydrated.
I think the biggest benefit to exercising outdoors is the mental aspect, it's nice to be outside, have changing scenery, enjoying fresh air, etc...however the type of training I do isn't conducive to being outside.