A meeting between Tim Cook and Steve Mollenkopf a year ago at Apple Inc.’s headquarters started with a tense moment.

The feuding leaders of two smartphone industry titans—Apple and Qualcomm Inc. —were there to discuss a long-simmering patent dispute. Mr. Mollenkopf, who suspected Apple of supporting a hostile takeover of his company, initially didn’t speak, leaving his general counsel to start talking, according to people familiar with the meeting.

The awkwardness punctuated a distant relationship between the chief executives that has turned their companies’ conflict into one of the ugliest corporate battles in history.

Apple has called Qualcomm a monopoly and said Mr. Mollenkopf has lied about settlement talks between the companies. Qualcomm has accused Apple of deceiving regulators around the world and stealing software to help a rival chip maker.

For two years, the companies have bickered over the royalties Apple pays to Qualcomm for its patents. Discord between the CEOs, who bring different management styles and principles to the table, has deepened the divide. They have dug into their positions as the dispute has escalated.