same here and me too are usually interchangeable. They mean that whatever the last person says applies to you, as well. So they may come after:
"I love pineapples" (I also love pineapples)
"I was held up in traffic" (I was also held up in traffic)
"I'll have a tea with milk, no sugar" (I'll have the same thing)
Me neither means the same thing, but is only used after negative sentences, such as:
"I don't like pineapples"
"I can't make it today"
"I haven't got any money"