Chances are no one's going to read this far down, but I'll post my 2 cents anyways.
I find that the stereotypical emo person listens to music that is depressing, will cut themselves, dress in black and colour their hair (usually red), and will wear darker colours, but usually have a band logo on it.
Now this I personally don't have a problem with, and I doubt many other people do either. It's a style, a label, a means of identification. If someone follows this trend while trying to find an identity, that's okay. Be it jock, goth, prep, or whatever, a look is a look, and will feel more comfortable than not having an identity.
That being said, I believe that the only reason to dislike emo people is the attitude. Now I know people who will dress in the emo style (although in some instances it's goth, and I don't know enough to tell the difference at a glance), and they have what I would call a normal attitude. Yes, they have problems, and they'll tell their friends about it. They then get support, and work past the problem.
That's fine. It's normal (in my opinion) to go overboard with emotions every once in a while. There's a lot of pressure and change during the teenage years (which is where emo people tend to show up) and outbursts should be expected.
It's when you can't let go of your issues that I have a problem with. If someone, anyone complains about their life but won't take any steps to change it, then I dislike them. I don't want them around me, and I certainly don't want to hear about what the latest issue was. If they aren't asking for help, but instead use their problems to generate attention, then I want to be far, far away from them.
This is the attitude of the stereotypical emo. This is the label that someone wears when they dress in the correct manner. I don't think anyone hates every person who dresses in dark clothes and skinny jeans, but rather the attitude that emo conveys. Knowing someone on the individual level is the only sure way to know what their attitude is, or even holding a basic conversation.
However it cannot be expected of us to do this. We stereotype and carry prejudices, which we use to gauge an encounter and prepare ourselves mentally. It's nearly impossible to approach a person as a blank slate, unless you forget all previous social contact each time. (Not something I would recommend).
To summarize, what is disliked is the stereotypical emo attitude. If you carry that attitude, regardless of how you dress or what music you listen to, then you are generally disliked. If you label yourself as a member of a social group, then you must expect that the stereotype will be pasted on you for every social encounter, regardless of whether you fit the stereotype or not.
Something to consider.