Perhaps it is the name given by English users when they first reached that region, but as we know after 1949, China adopted the pinyin and unified the spelling of English of names of place, Peking was changed into Beijing, something like Bombay of India into Mumbai, as the result of the nationalist efforts to make special identity. But Tibet is still in use, but not as Xizang. Why is it so? Hope it is not a political question.
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.
In this case it looks like it was taken from another language as Christophe Strobbe wrote.
Many times the English is taken from previous romanizations. For example "Peking" was an earlier romanization of 北京. Now English uses the more accurate "Beijing".