The Patton / Montgomery rivalry was enhanced and overblown by the 1970 movie Patton, staring George C Scott. Patton had many problems caused by the slapping incidents on Sicily and some ill advised comments in England, during the spring of 1944, for Patton to really force the issue with Montgomery. Patton also did not object very strongly to Montgomery's plans for Operation Husky. Nor did Patton object to Montgomery's plans for the Normandy campaign.
Montgomery also had more caustic relations with other American generals, particularly Omar Bradley. Monty does not come across very well in that movie. Montgomery had bad relations with Eisenhower, Bradley, Clark and several other lower level US Army generals. He said things that would have caused him to be sent home, if he was an American general saying such things to a British general. Montgomery's boss, Alan Brooke backed Montgomery to the hilt. Monty was also made into a hero by the British media. Eisenhower had to handle him with kid gloves, but he was very leery of giving Montgomery command authority over US Army soldiers by 1944.
Source: "There's a War to be Won: The US Army in WWII" by Geoffrey Perret