Charlie Chaplin and Stan Laurel. Laurel and Chaplin both performed in the same music hall company. Laurel got a reputation for imitating Chaplin almost perfectly, which didn't please Chaplin at all. Later Chaplin went into movies and became the most famous film star of all time. Laurel expected Chaplin to give him a phone call later on, but this never happened. As a result, Laurel had to struggle another 15 years before he made it big with partner Oliver Hardy. Chaplin mentioned Laurel only briefly in his autobiography.
MSNBC vs FOX News, for the title of the most loudmouthed/opinionated/rude/biased/partisan "news" network. With the most public fight being that between Keith Olbermann and Bill O'Reilly .
Got so bad that the bosses allegedly tried to make a truce, which didn't last long.
In the U.S., Ringling Bros. and Barnum & Bailey Circus versus Cirque du Soleil — the two biggest circus companies in the world, each with a distinctive approach to the form. Ringling is "traditional", with three tours that crisscross the country and are relaunched with new acts and themes every one-to-two years; Cirque is "contemporary", in that each show is a unique, theme-driven piece of theater that can run for years on end with few changes. Also, the former uses live animals and the latter does not, with the exception of some doves in Criss Angel Believe. Because Cirque shows are usually performed under tents in big cities and Ringling Bros. travels the broader arena circuit, this rivalry was largely one of style (though Ringling attempted a Cirque-style tour, Barnum's Kaleidoscape, at the end of The Nineties in addition to their traditional shows) until Cirque started adapting its older shows for arenas in 2007 to expand its audience.
Online magazines Slate and Salon can be considered this, having different stances, editorial positions despite both being generally liberal (though Salon leans considerably more into the left-wing). Both sites have also included among their ranks some major pundits and journalists (such as Christopher Hitchens for Slate and Glenn Greenwald for Salon), adding even more fuel to the rivalry.
The Polish computer game magazines Top Secret! and Secret Service. The latter was made up of the old Top Secret personnel who all went and left back in 1993 to start their own mag. These were the old days of videogame journalism, when they still could pepper their issues with very thinly veiled Take Thats at each other.
The United Kingdom and France (sometimes Germany too). Historically, England and France were like this a great deal, fighting a number of wars against each other all the way up to the 1800s. However, they've stopped since World War I broke out. These days, it's mostly sporting rivalries that the British media plays up to.
This creates a strange problem every year for fans of Michigan State University's sports teams: on one hand, Michigan State's biggest rival is the University of Michigan, but on the other hand, Michigan's great rival (as mentioned above) is Ohio State. The great majority of MSU students/fans are from Michigan, raised on a diet of hatred for everything related to Ohio. The result is that a perpetual debate in East Lansing is whether one should root for Ohio State or Michigan when they face each other. Consensus is somewhere around Henry Kissinger's observation about the Iran Iraq War: "Too bad they can't both lose."
The University of Oxford and The University of Cambridge, ever since the 13th-century, most famously manifests in the annual boat race between the two schools.
Similarly (if less old), The Royal Academy of Music and The Royal College of Music; unlike Oxbridge they don't have a specific contest, but expect at least one representative from each to reach the latter rounds of music competitions. Each institution has been known to jokingly refer to the other as That Other Place.
University College London (UCL) and King's College London also have a similar rivalry.
Across the ocean, there's Yale and Harvard in America and Queen's and McGill in Canada.
And MIT and Caltech.
University College Dublin and Trinity College Dublin have their own boat race, presumably in parody of the Oxbridge one. Both unis famously dislike each other, UCD being formerly 'Catholic University of Ireland' and Trinity being famously Protestant has an unfortunate role in this hatred.
Army and Navy.
University of Kentucky and University of Louisville. Sheer, undiluted hatred in the Bluegrass State.
Averted in the case of Wolfgang Amadeus Mozart vs. Antonio Salieri, who were friends in real life. The rivalry was made up by in a play by Pushkin long after the both men were dead.
Played straight with Georg Friedrich Handel and Domenico Scarlatti, two virtuoso megastars of the Baroque keyboard. It was later conceded that Handel was the more accomplished organist and Scarlatti the superior harpsichordist. Interestingly, Johann Sebastian Bach, the other obvious candidate for this trope knew Handel by reputation only; the two maestros famously never met all.
Maria Callas vs. Renata Tabaldi. Two polar opposite opera singers, they did manage to work together in the end.
During the Cold War the U.S.A and U.S.S.R. were each other rivals.
Benjamin Disraeli (Conservative) vs. William E. Gladstone (Liberal), considered to be one of the most famous of political rivalries, lasting a decade and a half. Two consummate politicians who loathed each other, clashed over the post of the British prime minister. (Gladstone served longer as PM).
Aaron Burr vs. Alexander Hamilton. Ended in a duel, Hamilton was killed, and the resulting public outcry destroyed Burr's political career.
Miyamoto Musashi vs. Sasaki Kojiro. Both were legendary swordsmen from the Warring States period of Japanese history. Their duel at Ganryujima, where Musashi killed Kojiro with a BFS fashioned from a boat's oar, is particularly famous.
Another famous rivalry from the same era was between Takeda Shingen, the Tiger of Kai, and Uesugi Kenshin, the Dragon of Echigo. While countless stories portray this rivalry as being one of the utmost respect between the two (even suggesting that the two were great friends when their nations were not at war), this probably originated sometime in the Edo period; Kenshin is recorded to have quite strongly hated Shingen.
Two Canadian Fathers of Conferederation, George Brown and John A Macdonald, were rivals. George Brown, the founder of the Globe and Mail, was much fiercer in the rivalry; his quest for power was ultimately fruitless (in other words, he was appointed a member of the Senate), whereas Macdonald was knighted and served as Prime Minister for 19 years.
Brown got a college named after him, though.
One of the reasons Brown didn't succeed, though, was that he wasn't given the chance: he was assassinated.
A number of leaders of the Liberal and Conservative parties of Canada fall into this trope, most famously John G. Diefenbaker (Conservative leader 1956-67, PM 1957-63) and Lester B. Pearson (Liberal leader 1958-68, PM 1963-68). The two men were extremely intelligent, accomplishished individuals who were able to be civil to one another.
Others include Sir Wilfrid Laurier and Sir Robert Borden, W.L. Mackenzie King and Arthur Meigen, Pierre Trudeau and Joe Clark, and John Turner and Brian Mulroney.
Pierre Trudeau (prime minister of Canada) vs. Rene Levesque (leader of Parti Quebecois, a mainstream political party advocating for the separation of Quebec). The duo fought bitterly over ten years over the independence of Quebec. The rivalry culminated during the drafting of the Canadian constitution, Trudeau, fearing the trouble Levesque is going to cause, had to trick Levesque by waiting until he was asleep before starting the backroom deals that was ultimately to produce the document. Levesque and the province of Quebec never forgave the trick, they still have not signed the constitution.
Joseph Stalin vs. Leon Trotsky. Most now agree that Trotsky would have made a better leader, but Stalin won the argument by having a ice axe stuck in Trotsky's head.
Appositely summarised by comedian Yakov Smirnoff thus:
The two most powerful men in Russia are Stalin and the last man who spoke to him.
Revisionist historians, working from newly released archival evidence, have argued that Stalin actually was the better leader, and that Trotsky lost because he simply was not as politically savvy. His account of Stalin and Stalin's rise to power is regarded as smear, insofar as it portrays him as a political non-entity of average ability who achieved power through sinister Obstructive Bureaucracy, and that the real Stalin was a multi-talented Magnificent Bastard.
* It's often joked, though with much substance, that the true bitterest enemies and rival Generals of World War Two, were General Patton and Field Marshal Montgomery.
There was a legend in the Army of the Potomac that McClellan and AP Hill had been rivals for the same girl and she had married McClellan. When they were hard pressed the soldiers said, "Why didn't you marry him?"
It's true. McClellan only won Mary's hand because her father forbade her from marrying Hill as he was a career officer. McClellan, on the other hand, was expected to retire from the army soon and dedicate himself to the administration of the family fortune and industrial businesses. Unfortunately he did not retire soon enough and made the Union Army suffer through his incompetence for a good while.
Otto von Bismarck vs Ludwig Windhorst (leader of the German Centre Party). Windhorst was by far the ablest and most dangerous critic of Bismarck's policy. The iron chancellor said once:"Everyone must have someone to love and someone to hate. I have my wife to love and Windthorst to hate."
Isaac Newton vs. Gottfried Leibniz. Clashed over who actually discovered calculus (now most agree that Newton came up with the idea first, but Leibniz probably came up with calculus independently from Newton). The rivalry was mostly one sided, with Newton and the UK treating Leibniz as an personal insult. This resulted in English mathematicians refusing to acknowledge developments in math from the rest of Europe, ultimately holding up English mathematics for about a century.
Thomas Edison vs. Nikola Tesla. Fought over the early electricity market, especially over whether power should be supplied as Direct (Edison) or Alternating (Tesla) current. Edison undertook a particularity vicious smear campaign against Tesla, which resulted in Tesla being pushed off public life and dying penniless and alone. Although Tesla has now been vindicated in that alternating current was better for transmission of electricity (if Edison had won, the electrical grid would have been so complex that electricity would probably never have replaced steam power as the driver of modern society).
Joseph Priestley vs. Antoine Lavoisier. One of the first ever rivalries over science, Priestley once sent a spy to steal Lavoisier's research on gases.
Richard Dawkins and Stephen J. Gould over the idea of gradual evolution or punctuated evolution.
Pioneer palaeontologists Othniel Charles Marsh and Edward Drinker Cope. In their case, it got ugly