Facebook Inc. Chief Executive Mark Zuckerberg has groused for years that Apple Inc. and its leader, Tim Cook, have too much sway over the social-media giant’s business. In 2018, his anger boiled over.

Facebook was embroiled in controversy over its data-collection practices. Mr. Cook piled on in a national television interview, saying his own company would never have found itself in such a jam. Mr. Zuckerberg shot back that Mr. Cook’s comments were “extremely glib” and “not at all aligned with the truth.”

In private, Mr. Zuckerberg was even harsher. “We need to inflict pain,” he told his team, for treating the company so poorly, according to people familiar with the exchange.

It wasn’t the first time—or the last—that Mr. Cook’s comments and actions would leave Mr. Zuckerberg seething and, at times, plotting to get back at Apple. The escalation of grievances erupted late last month in a rare public tit-for-tat between the two tech giants that laid bare the simmering animosity between their leaders, who exchanged jabs about privacy, app-tracking tools and, ultimately, their dueling visions about the future of the internet.

Apple has positioned itself as the protector of digital privacy, upholding a greater good, while often leveling criticisms at Facebook’s business model—without naming the company. All of that grates on Facebook, which sees Apple as overreaching in a way that threatens Facebook’s existence, and hypocritical, including by doing extensive business is China where privacy is scarce. A 2017 attempt to address tensions through a face-to-face meeting between the two CEOs resulted in a tense standoff.