Furore usually refers to a public uproar. Furor is the American and Canadian spelling of furore, and it has additional definitions it does not share with the primarily British furore. These are (1) violent anger, and (2) a state of intense excitement.
Some dictionaries erroneously list furore as simply a British variant of furor, and this inadequate claim is widely repeated on the web. But while furore and furor are indeed often variants of each other, furore is more narrowly defined. In U.S. and Canadian English, we might describe a rabid dog or a manically behaving person as being in a furor. This would sound odd to English speakers outside North America, who are used to the furore spelling and only the public uproar sense.
Furore refers to public uproar, often but not necessarily involving angry feelings—for example:
Meanwhile, bonuses are likely to be lower this year, although they are still likely to cause an inevitable furore as the rest of the country struggles. [Independent]
The furore comes just days after Rihanna took to her Twitter page to describe a racist incident in a Portugal hotel. [Daily Mail]
American and Canadian writers use furor for a public uproar—for example:
And so it has been with the furor that erupted in October 2010 over the mortgage industry’s dubious loan modification and foreclosure practices. [Washington Post]
The furor over the failure to bring the killers to justice led to a public inquiry. [National Post]
But furor can also refer to violent anger or a state of intense excitement—for example:
Brett Ratner resigned on Tuesday in a furor over an anti-gay slur and some nasty public sex talk. [New York Times]
Of course, fury would work in place of furor in contexts like these, which could explain why furor remains relatively rare in these senses.