The difference between theater and theatre is primarily one of spelling semantics. Speakers of British English are taught to use “theatre,” while speakers of American English usually use “theater.” The “-re” and “-er” difference is common to many other words in British and American English, like sabre/saber, center/centre, and so forth. Like many words which are spelled differently in British and American English, the words are sometimes used interchangeably, especially in America; the spelling “theatre” is more common in the American Northeast.
In some groups in the theatrical community in the US, people differentiate between live performances at a theatre and films displayed in a movie theater. Others choose to use “theatre” to refer to the performing art, while a building is a “theater.” These distinctions are not made by all writers, however, and there is no consistent rule for such usage. Usually people pick one spelling and stick with it.
Linguists often point to the tangled differences between the spellings and usages to illustrate the shifts that the English language has undergone over the centuries. Studies on historical usage of English in America and in Britain seem to suggest that spoken American English is actually closer to the “King's English” spoken when America was first colonized, according to Bill Bryson in Made In America, an exhaustive survey of American English published in 1994. The “-re” and “-er” is only one among many subtle differences between written English in Britain and the United States.
Many former British colonies retain British spellings for words, so people are more likely to see “theatre” than “theater” outside of the United States. American English may be so distinctly different because of America's relatively early independence, historically. Countries that remained under British control longer than the United States would naturally have continued to use British English, and their use of the language would have evolved with the British English-speaking community due to cultural exchange and formal written communications from the mother country.
Some people suggest that the difference between theater and theatre in the United States is one of affectation, suggesting that people who use the “-re” spelling are being snobby. Many of the arbiters of American English seem to prefer to use “theater.” The New York Times, for example, has a “Theater Section,” and many national theatrical organizations refer to themselves with “theater,” not “theatre.” Ultimately, the choice between spellings is up to the individual writer; they both sound the same, so unless there is a requirement to stick with a specific set of style guidelines, a writer can usually choose whichever version he prefers.