"Emo" is a term that has been so dramatically mutilated over the years the true definition is very hard to find.
Emo was originally a genre of music, emotionally driven and powerful lyrics. Usually listened to by ultra hip underground indie kids. Bands like Mineral , Christie Front Drive, and more well known Rites of Spring. Originating in the mid 1980's on the east coast, "Emocore"
was a offspring of the DC hardcore music scene.
I know nothing of Emo style, only I suppose its band followers emulating what band members wear.
Sunny Day Real estate is second wave emo. Like Fugazi, Which are bands circa mid to late 1990s that followed in the vain of original emo bands.