The United Kingdom of Great Britain and Northern Ireland (CIA World Factbook entry: http://www.cia.gov/cia/publications/factbook/geos/uk.html) is composed of the four countries England, Scotland, Wales and Northern Ireland - this is why we have the TLD .uk rather than .en, for instance. England is just the largest and most populous of these sub-countries. Given the violent and acrimonious history of these islands, a surefire way to annoy Scottish, Welsh, or Northern Irish people is to call them English. It also leaves out those of us who are some mixture of the four. 

Britain is the largest of the islands that make up the British Isles: the UK, England, Scotland etc are political entities. England and the others are called "countries", because in the past they've been separate nations, but they aren't any more: the only current nation-state (UN seat, etc) mentioned so far is the United Kingdom.

To make matters more confusing, the adjective used to describe a citizen of the UK is "British". This works neatly in Scotland, England and Wales, but could cause some difficulty in Northern Ireland: while Unionists would probably be happy to be called British, Nationalists would probably prefer to be described as Irish, as they'd rather that Northern Ireland be part of Eire than the UK.

Some British people are also English. Many are not. Most British people speak (British) English, though many only speak it as a second language - even leaving aside those whose families have immigrated recently, there are areas where the dominant language is Welsh or Scots Gaelic. Not even the law and currency are universal across the nation - Scottish law is different from English/Welsh law, and Scotland has different issuing banks from England (though the currency of one should be acceptable in the other). Northern Ireland also has some differences in law and its own issuing banks.

When in doubt, say British/Briton/Britain unless you're actually sure you mean English/Englishman/England.

Thanks to the members of London Perl Mongers for their many helpful comments on this entry.

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And in the name of proper EnglishLanguage, please use "DifferentFromNotDifferentTo".