Two scenes, two countries, half a decade apart, and two entirely different arguments about what music was for. The clothes came later.

Online the two words get used as though one were a milder version of the other. They name scenes that had almost nothing to do with each other: different countries, different decades, different music, different reasons for existing. The clothes converged much later, and mostly inside photographs.
The scene assembled itself around clubs rather than around a manifesto, and it took its sound from bands who were already working in a cold, spacious register.
It had addresses. The Batcave in Soho is the one usually named, and the naming of the scene itself happened in print about a year after it opened.
Emo started an ocean away and half a decade later, inside a hardcore punk scene that was arguing with itself about what the music was for. The answer some bands gave was that it could be about the person singing rather than about the state.
The word is short for emotional, and it began as something close to a criticism — a note that this band had turned inward. It was a description of subject matter, not of a wardrobe.
This is the part that gets lost. Both words named genres for years before they named a way of dressing. Goth described a sound built on reverb, low bass and theatre. Emo described songs that used direct personal language where hardcore had used slogans. Someone in 1985 could have been either without owning a single item of the clothing now associated with them.
Both were genres before they were wardrobes. The wardrobes are what survived.
Goth dress developed alongside the music and borrowed from much older sources.
The emo look most people picture is a different thing entirely, and it arrived roughly two decades after the music did — at the point the genre reached a mass audience.
Put the two descriptions side by side and the supposed overlap mostly disappears. One is antiquated and theatrical. The other is deliberately ordinary — thrift store clothes and a haircut.
Because the internet sorts by image, and both scenes look dark in a thumbnail. A photograph carries the black clothes and the eyeliner. It does not carry which records someone owns, which is the thing that actually separated them.
The confusion also runs in the other direction. Later internet aesthetics borrowed from both at once, took the eyeliner from one and the platforms from the other, and produced a look with no scene behind it at all. By the time those parts had been reassembled, the labels had stopped pointing at anything a record shop would recognise.
As descriptions of music history they are precise, well documented and easy to keep apart: Britain in the early eighties against Washington in the mid eighties. As descriptions of how a person looks in a photograph they are approximate at best, and neither one tells you what someone listens to.