There is no doubt that punk and goth subcultures play a very big part in the current emo style that many teenagers are sporting these days. Many describe emo fashion as taking the goth style (too much eyeliner, black hair with strips of vibrant color, skulls and crosses, you get the idea) and making it tighter and more cutsie (perhaps incorporating more manga hairstyles and Japanese youth paraphernalia like Hello Kitty)
Emo fashion also developed from a more nerdy somewhat indie and grunge look (like greasy dark hairstyle, your grandfathers old grey cardigan and his thick rimmed glasses, anything super old and moth eaten found in a second hand store...um just think of Weezer).
What is considered to be Emo today has really evolved to become something much more trendy and staged, think of all the "scene" queens and kings out there. But the emo look has really spread to a wide variety of youths who identify with each other for being outcasts and sensitive souls.