First, the verbs “to whine” and “to whinge” don’t connote exactly the same kind of behavior, any more than do the verbs “to whine” and “to bitch.” There are no true synonyms, language isn’t a code, and most common verbs refer to a range of actions, rather than something precise. Meaning emerges from usage; language is a negotiation, not a set of rules and definitions handed down from on high.
Second, there’s a repeated claim in the thread below that Americans who pick up Briticisms like “whinge” or “wanker” do so only in order to be “pretentious.” This is ridiculous. It’s 2007. British people—at least, British people who aren’t in a coma—are immersed in American vocubulary, slang, and idioms; and, increasingly, Americans are constantly exposed to British English. Unsurprisingly, this means that lots of Brits and Americans are picking up one another’s language quirks in a process of linguistic cross-fertilization exactly like the way language has always been transmitted throughout the entire history of the world. If you really want to insist that this makes those people “pretentious,” you’re a pinhead.