What is the difference between each other and one another?
Are there grammatical or context differences?
There’s no difference in grammar or usage and probably never has been.
The only reason you might have heard otherwise is that some pedants advocated restricting each other to two and one another to three or more.
Theodore Bernstein, a conservative pedant, said back in 1965 in *The Careful Writer *that while the distinction was common, there was no reason to enforce it other than convention. Good writers even then used each other for three or more and one another for two.
Fowler dismissed any distinction on the same basis in 1926, in Modern English Usage.
My rule of thumb is always that it Bernstein allowed it in 1965, any prohibition against it is ancient history today. If Fowler allowed it, then it was never an issue outside the minds of the illiterates who tried to prohibit prepositions at the ends of sentences.
Cool, thanks! I can move on to wondering about other things now. 
My impression is that it’s closely akin to the differentiation between between and among – which is not, specifically, two vs. three or more, but rather whether binary relationships and inferentially all possible ones, or group relationships, not necessarily all possible ones, is intended.
Though the terms are fairly synonymous, “The committee greeted each other” would to me imply that every member of the committee greeted every other member of the committee, one on one, while “the committee greeted one another” implies to me that there was a period of greeting in which the members of the committee greeted some other members, perhaps not every other member but the ones easily accessible to them. Likewise, “our cousins gave each other presents” means each cousin gave every other cousin a present, while “our cousins gave one another presents” may mean that or may mean they drew names and each gave a present to the cousin whose name he/she drew, or anything in between. And of course these are connotations – not necessarily absolute distinctions always valid.
Merriam-Webster’s take on it, basically buttressing Harpo’s contention.
I was very confused by this question until I saw other people’s examples because the situations where I often find the words “each other”, they mean something entirely different that “one another”. It is used as a synonym for “every other”, as in “As a member of the board of trustees, I support the measure, but each other member of the board does not.” Using “one another” there makes absolutely no sense. I’m not sure if the use of “every” vs. “each” has any rule or implication, but “each other” is used quite often on Magic: the Gathering cards when a card has an effect on “each other” card of a certain characteristic that the card itself happens to have.