Or if you want the cynical approach, we'll do the scorecard, listing the winner in each category.
Ankhs: Goth
Converse Shoes: Emo
Excessive Eyeshadow: Goth
Excessive Lyrics: Emo
Thick-leather Tomes: Goth
Thick-rimmed Eyeglasses: Emo
Delusions of Self-Vampireness: Goth
Delusions of Self-Enlightenment: Emo
Ear Piercings: Goth
Face Piercings: Emo
Marilyn Manson: Goth
Coheed and Cambria: Emo
Leftover Metal Fans: Goth
Leftover Swing Fans: Emo
Most Likely To Kill Self (And write bad poetry about it first): Goth
Most Likely To Kill Self (And write whiny songs about it first): Emo
I think that about covers it. Basically, emo is goth without the occultic symbolism. Conversely, Goth is pretty much emo without the conformity.
For further info one should, as always, check wikipedia.
http://en.wikipedia.org/wiki/Emohttp://en.wikipedia.org/wiki/GothEDIT: Just saw the most hilarious distinction. "They are a lot like Goths, but less Marilyn Manson and more Harry Potter."