nope. Goths came first, then later emos. Both wear black, have dark/black eyeshadow and nail polish, tend to dye their hair and style it in weird ways.
But if they have a lot of chains and piercings, or spikes in their clothes usually that's a goth.
If they write crappy poetry about their feelings and how everything is crap and nothing we do matters, thats probably an emo.
There's some crossover between the two. But a good rule of thumb is "if she hates herself, she's emo. if she hates the world she's goth"