When I was reading both magazines, Punch and the New Yorker shared many cartoonists, for example the late Bud Handelsman himself. One big difference was that New Yorker also published serious journalism and fiction and extremely long pieces, whereas Punch as an exclusively humor magazine had none of those things, but did have regular columnists and weekly features such as "Let's Parler Franglais!" I was a devoted reader of Punch in the Eighties and was extremely bummed out by its demise (not to mention one utterly horrific attempt at a revival).
Actually, there was a direct attempt to duplicate the New Yorker in England during the late Thirties with a magazine called Night and Day, featuring contributors such as Graham Greene, Evelyn Waugh, and Cyril Connolly. I've seen some issues and it really was a slavish imitation down to the layout and typography, but the actual writing wasn't nearly as good. Its sole claim to recognition today is the magazine's demise in a once-notorious libel suit: