Why is 西藏 Tibet but not Xizang in English as are other names of place in China?
The name 西藏 / Xīzàng is fairly recent. 

Both Wikipedia and the Encyclopedia Britannica agree that the name "Tibet" is derived from a different language than Chinese, probably Persian "Tubbat", but that may not be the original source of the name (Wikipedia). Britannica says:


  The name Tibet is derived from the Mongolian Thubet, the Chinese
  Tufan, the Tai Thibet, and the Arabic Tubbat.


Dr. Elliot Sperling wrote that " in discussing Tibet as an integral historical unit the name is clearly of limited utility and can create confusion" and that the Chinese edition of the book Authenticating Tibet therefore uses a different name, namely "Tubote" (a name closer to the non-Chinese origin of the name "Tibet"): 


  Indeed, it is due to the specific limitations of that term that it was
  decided that the Chinese edition of Authenticating Tibet would employ
  the term “Tubote” 圖伯特, a name that is used now by a growing number of
  writers and bloggers on both sides of the Taiwan Straits. As a result,
  I prepared the following introductory essay for the Chinese-language
  edition of Authenticating Tibet in order to explain the choice while
  at the same time pointing out that the term “Tubote”is not a recent
  concoction: it has a history as a recognized Chinese-language
  designation for Tibet.


His essay Tubote, Tibet, and the Power of Naming also discusses the etymology of 西藏 / Xīzàng.

The origin of the name "Peking" instead of Beijing is a different story. European commercial travellers to China often came to the country through ports in south-east China, where dialects other than Mandarin are spoken. For this reason, some people think that the name "Peking" derives from one of those dialects, most likely Cantonese, where the pronunciation may have sounded like [pakkiŋ]. An alternative theory is that the European name was borrowed from Mandarin before the shift in pronunciation from [k] to [ʧ]. See Bill Poser's article BEIJING, PEKING, PEIPING AND ALL THAT on Language Log.