In standard English, these are two different words! It is highly misleading of any dictionary to list a misspelling as a permissible variant. In my opinion this is not a matter of AE/BE but of correct English usage.
'Premier' is an adjective. It means leading, best, first in quality or rank: the premier product in its field. Another example: the premier attraction, meaning the principal or main thing people want to see, the most important feature. If you spell the word with an E at the end in this sense, it is wrong, period, end of story. (Well, there is one exception: namely, when it is a French feminine, e.g., premiere danseuse.)
'Premier' can also be a noun, in which case it means a political leader, like a prime minister: the premier of the Soviet Union.
'Premiere' (or 'première') is a noun. It means a new theatrical production, the first time something is shown to the public: There was a red carpet for the actors at the premiere of the new film.
In this sense 'premiere' can also be used as an adjective, but only with the same meaning, namely first in chronology: The premiere production of the play was done in the round, but subsequent productions have mostly been on proscenium stages.
'Premiere' can also be a verb: The ballet premiered in 1933 and was revived in 1970. Some purists initially objected to this verbing of a noun, but it serves a useful purpose and is now completely accepted.
Good dictionaries should at least have a usage note explaining the difference. It's unfortunate that several don't seem to, and some even have incorrect examples, but that's no excuse for perpetuating the confusion in LEO.