I was looking at both the english and spanish Wikipedia pages for Augustus Caesar and I noticed that in the english one he is called by the name I know him as but in the spanish one he has a spanish derivative of this name, César Augusto. I've also seen this in the names of saints (Saint Peter and San Pedro, for example).
Why do we do this? If we were to refer to someone more current like Raúl Castro or even Niccolo Machiavelli we don't use an English version. Why don't we refer to these people by the names they would have been called when they were alive?
As for the saints, it is very subjective as to use their native name and spelling or the anglicization. For example Saint John of the Cross's real name was San Juan de Yepes or San Juan de la Cruz. But it is unheard of to call Mexico's patron saint Juan Diego John Diego. So the why really comes down to a case by case basis often on cultural heritage etc.
As for Chinese, which you will often still see spellings like Mao Tse Tung or Chiang Kaishek. A lot of organizations will fall back on the past Wade Giles system rather than Pinyin. But because many people are unfamiliar with the correct pronunciation of the Wade Giles system you get Jiang become "Chang" and basically changing the persons name.