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Escape from New York (1981) More at IMDbPro »

The most logical reason is that he did it in retaliation - he felt cheated by Hauk because he was not told about the explosive charges in his neck until AFTER he agreed to take on the mission and the injection and thereby endangering his life. This is the same as his motive for doing a similar thing in Escape From L.A. (1996) when he shuts down the Earth.

Another reason could be that Snake broke the tape out of respect for those who helped him escape (Cabby, Maggie, Brain). Keep in mind, the President (Donald Pleasance) wasn't too appreciative of the others (saying they did a "service to their country") and was more worried about how he was going to look on TV to address the summit leaders.

And of course, it could be that he really doesn't give a shit.

The contents of the tape are never explicitly revealed in the movie, but we can make certain assumptions.

Hauk states that the United States is at war, and that the President was on his way to a summit called the Hartford Summit. China and the Soviet Union are waiting at the summit, so the implication is that (in the movie's alternate future) the U.S. is at war with them. Hauk says that the President's arrival is "critical to the fate of the human race," so the summit is most likely a last-ditch effort to put an end to World War III, an all-out nuclear war. Without the President and the tape, Hauk says, the representatives of China and the Soviet Union will go back to their countries, presumably to continue fighting the war.

One possibility is that the tape contains the speech that the President was going to deliver at the Summit. That seems unlikely, since there's no reason the President couldn't have improvised a new speech containing the basic theme of the old one. Besides, the President could have written the speech down instead of recording it on a tape, and his staff would have had a copy of the speech.

When asked about the tape, Hauk asks Snake what he knows about nuclear fusion. At one point in the cab, a brief portion of the tape is played. The voice on the tape describes radioactive elements before it's shut off. So we know the contents of the tape are related to nuclear technology.

This leaves two possibilities. One is that the tape's recording contains a description of some revolutionary form of nuclear power which the President is sharing with the Communist nations. Giving the technology to the enemy might be a goodwill gesture or revealing the technology might have been a condition of Russia and China's appearance at the summit. Another possibility might be that the tape describes all of the nuclear secrets of the United States. Revealing these secrets might be (again) a goodwill gesture to the enemy or part of the terms of the summit. Either way, the President failing to deliver the technology ends all hope of the summits success.

But the boring answer is that it really doesn't matter what was on the tape in the briefcase, as the tape and the briefcase is a MacGuffin, a plot device to get the plot going.