What exactly is the difference between anniversary and commemoration? [closed]

I think the important phrase is act of, in the definition of "commemoration". An anniversary occurs without any active involvement. The anniversary of my wedding occurs on the same date each year, whether or not I buy my wife a gift. We might commemorate the anniversary by going out to dinner and exchanging gifts.

The example of Tiananmen Square is interesting, since China historically is adamant about not commemorating the anniversary of the event:

Several people have been arrested, or at least taken away for questioning, for attempting to mourn the victims publicly. One man was questioned for wearing a button that had the V-for-Victory sign and the word "Victory" on it in 1990. According to the New York Times, another man, in 1992, named Wang Wanxin "was dragged away after he tried to unfurl a banner calling on Deng Xiaoping [...] to apologize for the 1989 army crackdown". Some other modes of commemoration included 50 dissidents staging a 24-hour hunger strike in 2000 and private memorial services in people's houses. In 1999, Su Bingxian lit a candle for her son who was killed in the massacre, while others lit ten symbolic candles.

In short: the anniversary simply occurs a set number of years after the event. An anniversary may or may not be commemorated (celebrated).