First of all, I think the Argentinian people should thank the Brits and particularly Thatcher for getting rid of a truly heinous junta that ruled the country with an iron hand. Democracy came back to Argentina much earlier than it ever would have.
Second, if there was no oil and gas involved, I don't think there would be as much passion from the government, although I think the Argentine populace truly thinks the islands belong to them. But the Islanders themselves don't want to be part of Argentina.
This may be a stretch, but I think there is a parallel with the Spanish American War in 1898. Although we did not "seize" Cuba, we did invade on what turned out to be false pretenses. If we had claimed the Island, how would that have looked today? The War made the US a world power, but we acknowledge today that maybe it was not a moral fight. Much of the language in the newspapers of 1898 was similiar to what you can read in the Argentine press today. American were outraged that Spain still had "colonies" in the Western Hemisphere. Argentina is similiarly outraged that Britain has territory so close to their shores.
History repeats itself, and lives are lost. Great question.