The difference is the fact 'Me either' isn't technically correct. It's a colloquial phrase that means 'me neither'.

In written English, it should be "Neither do/would/have/will I," but orally, most Americans say "me neither" to mean "not either." It is a way to agree with someone else about something that is stated in the negative. Some Americans say "me either." to mean the same thing, but it's a minority. We'd understand either one to mean the same thing. Example: A: I'd never eat chocolate right before bed. B: Me, neither. (or neither would I) A: I've never been to Indonesia. B: Me, neither. (or Neither have I) A: I don't like cold weather. B: Me, neither. (or Neither do I) A: I won't go home for the holiday B: Me, neither (or Neither will I)