Not really. You may see people argue about this but they are really used in the same way. You'll even hear common phrases like: Blah blah blah... each and everyone of you. Each and everyone of you is probably one of the most redundant phrases in speeches. All of you = everyone = everybody = each one of you

Each other - 2 people or things. One another - more than 2 people or things. e.g. Harry and Ann looked at each other. Mary, Shelly and Percy fight with one another.

I agree with Oscar. The meaning is the same. The only real difference is that "each other" is more common, especially in speech. People rarely say "one another" in conversation. In writing, however, you'll see both phrases. The distinction Tanya makes is not correct -- both "each other" and "one another" can refer to two people or more than two people. Her sentences are both fine, but you can also reverse them and say "Harry and Ann looked at one another," or "Mary, Shelley, and Percy fight with each other."