Genres are just artificial constructs, and we sometimes get distracted in this thread debating whether you consider a band 'emo' or not. I think most would say Simple Plan is firmly in the pop-punk, where Senses Fail is probably closer to being what I would consider emo, but still would consider pop-punk. But, it has nothing to do with how 'dark' or 'heavy' something is. Saves the Day is like quintessentially emo and I think many would consider them firmly pop punk. It's the highs and lows, man. Moody music; minor scale verses into major scale choruses and moody tonal shifts.
Take this song, for example:
Rocks Tonic Juice Magic
This song seems like just exploitive try-hard violent, but if you pay attention, the desperate violence action slowly cranks down and he slowly becomes more sincere and vulnerable. In a song about mutilation and cannibalism, the 3rd verse about buying Lemonade comes off as ESPECIALLY cruel, because it's not just pornographic B-Horror Movie violence... it's actually seemingly real and vulnerable. He starts out being desperately angry, to feeling betrayed, to feeling bitter-sweet about how things were at the end. To me, this is Emo. So, it's all up to how you feel about it. Brand New may not be so emo anymore, but they did release one of the best mass-appeal emo records ever IMO.