The distinction is not a strict rule. According the this and this Wiki, both spelling can be used to denote either the country or the ethnic group. A quick search for both terms will show that they can be used interchangeably. It is good practice, however, to standardize the spelling within a single document so as not to bring about unnecessary confusion. I would not recommend making the differentiation in formal writing, unless you have something to say about it. (Such as, needing a shorthand to differentiate between Uzbeks who are Chinese nationals and those who are not.)
Chinese has its own convention for transliterating proper nouns from a foreign language. As long as the convention (primarily phonetic, often also lyrical) is followed and people can easily see what you mean from the context, you will be doing fine.
The Uzbel Pass, for example, has been rendered as「烏仔別里」、「烏茲別里」and「烏斯別里」in the《西北邊界地名譯漢考》and《清續文獻通考》published respectively in the late Qing and early Republican period, less than two decades apart. These are encyclopedic documents which record both historical and contemporary spellings of the term, so you get an idea of how the language works over here.
The preference for「孜」in place of「茲」to name the people within Chinese administrative region in this case is likely to be based on lyrical and/or practical grounds rather than brute politics. The latter is usually used as a preposition or pronoun to mean this, here or now in formal writing. On the other hand, the former has the unambiguous positive meanings of diligence and meticulousness. Which, incidentally, makes the differentiation a good way of making distinctions between the specific (Chinese nationals) and the general (Uzbeks in general) .
@user6065 has provided a very good lead. 「茲」is actually a heteronym with an alternative pronunciation of ci2, which is commonly used in foreign place names like 「龜茲國」(qiu1 ci2 guo2, also written as 「兹」, to show its affiliation with 「慈」). The point of contention over here is that the Uzbeks call themselves Oʻzbek, with a voiced and inaspirated consonant 'z' (sounds like 'Ooh-ze-bek') in the middle, while the Russians call them узбек (sounds like 'Uh-s-bek') , with an unvoiced [s] sound in the middle. The [s] is deemed to be aspirated, like ci2. (It is not; aspirated [s] should sound like the front 'S' in 'States'.) Therefore the choice of 「孜」 in place of 「茲」 is deliberately made as a show of amity: by calling the Uzbeks as they call themselves, rather than as the Russians call them. In fact, the phonemic contrast in question over here between Oʻzbek and узбек is actually that of the voiced [z] and unvoiced [s] rather than the aspirated [tsh] and unaspirated [ts] (between ci2 and zi1). Moreover, 「兹」(ci2) is historically pronounced with the voiced consonant [z], like Oʻzbek. So there actually is a more erudite way (traditionally adopted by the Chinese) to express friendship - that is, to be loyal to the original transliteration, based on middle Chinese pronunciations and read it as it is.
@hinen, check this out:
As pointed out by @NanningYouth, China was still under the Nationalist Government in 1934. Xinjiang under Sheng's rule nonetheless refused to kowtow to the Nationalist government till as late as 1943. Even so, his submission did not last long. Moreover, Sheng suggested in 1941 that Xinjiang should join the Soviet Union and relinquish any relationship with China. This shows that Sheng is more likely to be pro-Russian than pro-China (Sheng was ethnically Manchu, not Chinese), and whatever considerations he had when he proposed a differentiation of the term, it was definitely not in service of the Chinese government at that time.
The PRC government of today is merely an (possibly passive) inheritor of whatever custom that's been left behind. There's not doubt they may try to tweak the situation to their own advantage, but till now I've seen no concrete evidence that this has been done, as far as the naming of the ethnic group is concerned.
Given the above, here's a summarized conjecture of the possible political motivation (if any) behind Sheng's choice when he proposed to make that differentiation:
- We know that 「茲」can be pronounced as zi1 or ci2, with zi1 closer to the s in the Russian name, узбек, (don't get confused with the 'z' in the hanyupinyin) and ci2 closer to the voiced [z] in the Uzbek name, Oʻzbek.（Contrary to what is proclaimed in this wiki.) Some Chinese dialect still preserve the voiced「兹」ci2 today, such as the Shanghainese Wu dialect, which reads「兹」as [zɦ].
- 「孜」shares with「茲」the former pronunciation.
- Given that「茲」can be pronounced both the Russian way AND the Uzbek way, it is highly possible that Sheng chose「孜」to deliberately pick the Russian way of pronouncing 'Uzbek'. (Our current English pronunciation of 'Uzbek' is also closer to the Russian reading.) That Sheng was pro-Russia made this even more possible.
All said, the above 'political motivation' remains a conjecture and awaits more evidence to prove if it is true or not.