The Phillies most intense rivalry in recent years is the New York Mets. They also have rivalries with the Florida Marlins, Atlanta Braves, and somewhat of a rivalry with the Pittsburgh Pirates. The rivalry between the Pirates has been diminished over the years due to the poor play of the Pirates and realigning of divisions.
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Depending on who you categorize as a rival, the Yankees have many. But the most prominent rival is the Boston Red Sox. Their infamous rivalry began in 1918, when the acclaimed Curse of the Bambino started. Harry Frazee, owner of the ballclub at the time, offered Babe Ruth to the Yankees in exchange for $100,000 so that he could fund his new Broadway production. Babe Ruth's legacy soon unfolded in Pinstripes. There are also the cross-town rival Mets, who reside in Queens. Besides them, only division rivals Tampa Bay Rays, Baltimore Orioles, and Toronto Blue Jays stand out as top Yankee "rivalries".
THE rivalry began over 100 years ago. Ever since, it has taken on a life of its own! It starts with the owners, the front office, the coaching staff, the players, right down to the their fans, (past & present). Unlike some other rivalries, these two teams have never re-located, to, or from another city since their inception into the league. With the exception of numbers being added to the jersey's back, their basic team uniforms have remained the same since before the Titanic sunk!
I feel no other rivalry can match this historic, Boston Red Sox - New York Yankee rivalry.
Especially when in 2004, the Red Sox made MLB playoff history as the first team ever to comeback from a 0-3 games deficit to win the American League Championship Series. Oh, BTW, their opponent was the New York Yankees. That historic comeback enabled them to go on to win their first World Series in 86 years by sweeping the Cardinals. The Red Sox followed up their 2004 Championship with World Series titles in 2007 and 2013.
To me, this rivalry is not only the best in MLB, but in all American pro sports.
Phillies are their divisional rivals, but Yankees are their city rivals who they fight with for fans, sponsorship etc.
The Braves, the Yankees, the Phillies, the Nationals want to be, but the Mets are to good for them.