The Canon 80d is a hobbist camera. By all accounts it is a pretty good one. Not particularly good at sports, and not for high end video.
The Nikon d500 is a pro grade camera. It is more rugged, better weather sealing, better auto-focus system. The video is ok, but it requires manual focus for moving subjects. Most professionals use manual focus for video so they can control DOF, specific focus point and so on.
Neither camera would be my choice for either of the things you said. For professional photography I would go with a Nikon d850 (if cost were not a problem) or a Nikon d750 if you wanted to save a little.
For video I would go with a Panasonic gh4 or gh5. I would also consider a Sony a7s II.
What I use is a Gh3 for event videos. It can record very long (unlike every other DSLR type a Panasonic GH can record more than 29.5 minutes).
For out (especially rainy weather) I use a Sony rx10 for video. I went with the gh3 over the then just released gh4 because I tried both and decided I didn't want to deal with 4k video files.
For sports I mostly use a Nikon d750, or occasionally an older d700 with a grip which boosts FPS very high.
I would get a Canon d80 if I were in to casual family photography 80% and casual home movies 20%.
I would get the Nikon d500 if I were in to sports or wildlife photography at least 50% of the time. (the other 50% could be anything) and I only needed to blog style videos or video focus speed didn't matter.
Other wise I would figure out which type of imaging I care the most about, and gear towards that.