What happens when I display 1080p video on a x720 resolution setting?

Although from a technical point of view, people would say you caot see the difference, I have found this to be incorrect if you use Youtube as example.

You can indeed see the difference, but its very hard. A 720p quality is already good, but because of the scaling down from 1080p to 720p on the fly, it is possible to see a quality increase. This is mainly due to how movies are compressed at youtube. Youtube's attempt to trim video's from 1080p to 720p remove certain details that your monitor doesn't. This is solely a compression thingy though, but it can be seen. Even better way to see the difference is to play a QuadHD: 4096p video on a 1080p screen and compare it to the 1080p equivalent.

If you keep youtube out of the loop and look at movies specifically compressed to these formats without any data loss, technically speaking, you won't see a difference.