Emo is a complex beast. It came from the late 80s in the form of Rites of Spring, and slowly changed. From there it became Sunny Day Real Estate, Jimmy Eat World, The Get Up Kids, and Cap'n Jazz. But as the remaining ones of these bands slowly changed the style of their music, they were still associated with the term and bands that were like them got the name. Fall Out Boy could be considered emo because of the fact that Pete Wentz admitted something along the lines of if there wasn't The Get Up Kids, there would be no Fall Out Boy. But The Get Up Kids still are not anything like old, decent emo. It's just that genres change, they move on.
But now people who listen to this new MTV emo adopted a fake personality. Make up, crying, pretending to be deep, wallowing in pain and crying because no one accepts them. They're annoying and fake. They are what most people call emo now, and they tainted the good name of Rites of Spring.
Emo: WAAAH MY GIRLFRIEND BROKE UP WITH ME! Time to cry while listening to Dashboard Confessional and cutting my wrist. The world doesn't understand me!