First, is its intersection with the internet. The genre cannot be understood without myspace and limewire and the like. It's the sound of that time.
Second, is how (like grunge) it was a media phenomenon that tried to unified very different bands. My Chemical Romance, Fall Out Boy, and Taking Back Sunday are all labeled "emo" but sound nothing alike. But they did brush elbows and were in the same place at the same time. Taking Back Sunday even did a stint opening for MCR in their return tour.