While the recently popular term Brit is making some inroads into the use of these two words in informal contexts, both Briton and Britisher are in reputable use for "a native of England, Great Britain, or the United Kingdom"—and they are sometimes taken to include people from the Commonwealth nations—often in contrast to American.
Briton, the OED tells us, was extremely popular in the 18th century; its use seems to have receded somewhat in the late 19th and early 20th centuries but to have come back a bit since then. It is more used in the U.K. than in the U.S.:
... we Britons by our situation —James Harris, Hermes, 1751
... I had the inalienable right of a freeborn Briton to make a morning call —Lewis Carroll, letter, 11 May 1859
It was deep in Alabama that a fellow Briton persuaded me to change my ways —Vincent Mul-chrone, Punch, 2 June 1976
A mori poll taken in Britain in February found that, although two-thirds of Britons said they liked Americans, over half did not trust Mr. Reagan's judgment —The Economist, 26 Apr. 1986
... difference between the speech of Americans and Britons—Baron 1982
Britisher is more common in American than in British English:
... I had become used to being the only Britisher permanently established in the region —R. C. H. Sweeney, Grappling With a Griffon, 1969
... it is naturally favored by Britishers on holiday — Town & Country, January 1983
... no more intriguing... than any number of other pleasant Britishers might have been —Ethan Mord-den, NY. Times Book Rev., 23 Dec. 1984
"I am a Britisher," May cheerfully tells everyone in the accent of her native Yorkshire —Giovanna Breu, People, 2 July 1984
While both of these terms appear to be in current good use, both have been and perhaps still are involved in some controversy:
... are we Britons or Britishers? I don't like either very much —Hardcastle, Punch, 20 May 1975
Nobody likes to be called a Briton, though it saves space in journalism.... Britisher is an Americanism —Longman 1984
... we beg him to avoid in future the odious and meaningless word "Britisher" —Anthony Powell, Punch, 8 July 1953
The OED says that Britisher is apparently of American origin (as does the Dictionary of American English)—a point disputed by Richard Grant White. White in turn was disputed by Fitzedward Hall. Brander Matthews in Americanisms and Briticisms (1892) pointed out that none of the authors cited under Britisher in the OED were American. Nevertheless, Americans did use the term:
... to excuse the curiosity of a Britisher —William Dean Howells, The Lady of the Aroostook, 1879
White's contention, apparently supported by Matthews, seems to embody a view shared by contemporary dictionary editors; Fowler 1926 notes that many of them attach pejorative or other warning labels to the term. Fowler, however, doubted (correctly it appears) the accuracy of the labels in American dictionaries, and commented that if Britisher was used by Americans in reference to the English, the English had no right to object. The controversy seems to have subsided somewhat since then, except for a few flare-ups like those quoted above.
Briton seems to have hit its low point shortly after World War I. The OED says that in the late 19th century it was used chiefly in poetical and rhetorical contexts, and Henry Bradley in The Society For Pure English's S.P.E. Tract XIV(\923) says that it suffered from its use in the 18th-century patriotic song "Rule Britannia." From this bottom it appears to have rebounded.
It seems possible that the lingering memories of these old controversies may partly explain the recent popularity of the more informal and sometimes derogatory Brit. (资料出处：韦伯斯特英语用法词典)