Well you see Casey's last band, The Receiving End of Sirens, was considered post-hardcore, and in many ways, the first two albums were aesthetically similar to the music he produced with them. Also, for contemporaries they're very often lumped together with the likes of Panic! and Forgive Durden, who are what I'd call "artsy emo."
With Act III, I feel like there wasn't a huge departure from the TDH sound, except that the TDH sound no longer meant that it sounded hip and poppy and like something an emo kid would listen to (I don't mean to generalize, but it's all about the point I'm trying to make). Act II was an angsty teenager's album with lyrics about love and heartache; Act III has a complex and intricate storyline well beyond that.
I feel like I'm not really getting my point across, but what I'm basically saying is Act III was the step from making music for young people with a message they can relate to, to making music with serious intentions. I don't know, I hope you get it. Listen to Act II and III back-to-back if I'm not explaining it right.