Emo is called emo because of the lyrical content and context of the song. It's not defined by how the music actually sounds. With that said, though, emo does branch off based purely on how it sounds. For example, screamo. Stuff like Bullet For My Valentine and, though I hate to say it because I love them, Funeral For A Friend. Alkaline Trio have very few songs with 'emo' lyrics, but their sound is very similar to blink-182, who are pop punk, so I've always categorised Alkaline Trio under that genre.
What's everyone stance on pop punk, anyway?
Emo definitely has a sound, obviously the lyrics are a big part, but it definitely has it's own sound, if we use that definition then every band ever is probably emo. I don't know if you're saying BFMV and FFAF are screamo bands but they're really not! Saetia are usually the first band mentioned when people talk about screamo, their retrospective is probably on Spotify/YouTube so check it out if you wish.
I think people are getting confused because all of that early/mid 2000's stuff hyped up by the media is not really emo, it's probably more pop punk/post-hardcore/whatever but Kerrang! and AP decided to call it emo and it stuck which is unfortunate because it's not really the same thing at all. The only reason bands of that ilk get called emo is because what was fast becoming a popular and trendy scene at the time needed a name, I don't know if you've ever read Kerrang! but they don't have a clue about anything so I certainly wouldn't take their word for it.
The first post in this thread could do with updating because there's only Plaids and Human Hands that hit the mark.
I mean, listen to stuff like Mineral, Christie Front Drive, Sunny Day Real Estate, The Blue Period, Empire! Empire! (I was a lonely estate) and then listen to My Chemical Romance, Fall Out Boy, The Used etc. it really isn't the same thing.
Pop Punk is not really my thing, there's a few bands I like though. The latest Lifetime record was more Pop Punk than their previous stuff and I really enjoyed it.
Edit: Sorry if I come off as an asshole but that's really not my intention, it's one of my favorite genres of music and I think it's awesome that we can discuss it in this thread