Like many friends from different orbits, Vice President JD Vance and Britain’s foreign minister, David Lammy, got together on Friday to bond over a hobby — fishing — before sitting down to hash out their differences, in their case, how the United States and Britain should respond to the deepening humanitarian crisis in Gaza. Mr. Vance, a Republican whose public positions rarely diverge from President Trump’s, acknowledged there was daylight between him and Mr. Lammy, a Labour Party minister under Prime Minister Keir Starmer, starting with Britain’s announcement last week that it would recognize the state of Palestine unless Israel agreed to a cease-fire with Hamas. “We have no plans to recognize a Palestinian state,” Mr. Vance told reporters as he met Mr. Lammy at the foreign secretary’s grand official country residence, Chevening House, southeast of London. “I don’t know what it would really mean to recognize a Palestinian state, given the lack of a functional government there.”Mr. Vance said Mr. Trump was sticking to his goal of making sure Hamas is never again able to strike Israeli civilians as it did on Oct. 7, 2023, though he said the president had been moved by the “terrible images” from Gaza and wanted to ease the suffering.