In most contexts, there is no difference in meaning between theater and theatre. Neither has any special definitions in general usage. The main thing that most English speakers and learners need to know is that theater is the preferred spelling in American English, and theatre is preferred virtually everywhere else.
Some Americans do make distinctions—for instance, that a theater is a venue while theatre is an art form, or that a theater is a movie theater while a theatre is a drama venue. There is nothing wrong with making these distinctions, but they are not consistently borne out in general usage. Even in 21st-century writing on the art of theater, the more American spelling now appears for all senses of the word.
The American preference for theater is a late-20th-century development (though the spelling itself is a centuries-old variant), so it is understandable that some people still resist it, and its newness means that exceptions are very easily found, but in this century the preference is entrenched. Searching a selection of 40 American news and cultural publications that put their content online, theaters appears 8,500 times from 2000 to the present, against just under 200 instances of theatres. This just suggests that theater is the preferred spelling for actual venues (the art form is a mass noun so would only rarely be pluralized), which no one seems to dispute. What’s more interesting is that the phrase theater critic appears 260 times against three instances of theatre critic, theater actor appears 43 times against zero instances of theatre actor, theater scene appears 60 times against two instances of theatre scene, and the phrase contemporary theater appears 27 times against two instances of contemporary theatre (and both of these are in names of buildings).
Meanwhile, ngrams meant to draw out any preference for theatre as the spelling for the art form in U.S. books shows theater ahead even in these uses over the last few decades (the links in this sentence point to some of the ngrams).
He rails against naturalistic theater as trite and trivial. [Marin Independent-Journal]
First, historians of theater and drama have described an increasingly out professional class of theater practitioners. [Cast Out: Queer Lives in Theater, Robin Bernstein]
The theater scene is buzzing this week, with a head-spinning array of recommended shows running on local stages. [Buffalo News]
Similarly, the field of theater studies to date has largely ignored her dramatic output. [Mama Dada: Gertrude Stein's Theater, Sarah Bay-Cheng]
Outside the U.S.
Perhaps it’s forehead-slappingly obvious to say so, but theatre struggles to do likewise. [Guardian]
In the spring of 1968 Ruben moved to Canada with the intention of serving as the Toronto Star‘s backup theatre critic. [Establishing our Boundaries: English-Canadian Theatre Criticism, Anton Wagner]
Budding theatre stars in Chester are invited to take part in a number of classic West End themed workshops. [Chester First (link now dead)]
She … applauds parents for taking their children to the theatre when there are so many other entertainment options. [Herald Sun]
Theatre in U.S. building names
Theatre often appears in the proper names of American theaters—for example, the American Ballet Theatre, the Muncie Civic Theatre, and the Genessee Theatre. When referring to one of these establishments by the common noun, theater is the usual spelling (in the U.S.)—for example:
Denver loses another theater space when the Vintage Theatre is bounced from its home at 17th Avenue and Vine Street. [Denver Post]