On a day of G.O.P. intrigue in Washington, Trump and Pence strike sharply different tones.

Mr. Trump and Mr. Pence had been in somewhat regular contact after leaving office, speaking several times by phone in conversations that avoided discussion of the Capitol riot, according to their advisers. In an interview last year, Mr. Trump said that he had never told Mr. Pence he was sorry for not acting quicker to stop the attack — and that Mr. Pence had never asked for an apology.

But a rivalry has flared up behind the scenes.

On Monday,…