In Europe, there are distinct cultural differences down through history, from the Barbarians and Greco-Romans of ancient times, to the Christian Kingdoms, to the secular, democratic modern era (simplified, of course). How different would life in the dynasties of China have been? Would someone from Han China recognize the China of the Tang? Of the Song? The Qing? What differences would have been most apparent?
That question has many implications and covers a long range and several complex topics. First of all it is implied in your question, that there is a distinct history of China, but if you look at the different dynasties and the territory and the "cultures" in their territories they contain, it is noteworthy there, where huge differences between some of the dynasties and their territories as.
Therefore we always need to see how much of Chinese history is affected by political concernes of the historians who wrote the official dynastic histories, that still have a notworthy impact on the dynastic historic periodization we find often when talking about Chinese history.
Not to be to much focused on the writing, as I already talked about the writing of history, but in general what is very special for China is of course their language and their writing, their are definitly developments in the spoken language that would not impact the character based script AND the very traditionalist court with its official writings. Therefore, when you learn Mandarin today you get told what classic Chinese is and there are definitly developments in it that are very significant but you still have to expect to find a high level of conservative attitudes in the language that was used in written form until the 19th or even 20th century [and in case of some writeres even beyond], that go back to classic Chinese of the time of Confucius. Our teacher likes to tell us repeatedly that the Chinese of that time that was used for writing was a spoken language, while at later times around Tang, we would see a shift between the official writting language of the court and the language that was actually spoken. It is similarly important to keep in mind that there are dialects until today in China which all have siginficant influences until today. Accordingly still today it is not 100% clear what you mean when you say "Chinese" to address the language, which always leads you to make some more qualifications what you are refering to.
Another big topic to cover could be seen of course in the usually attributed changes between the dynasties that are foreign and those that are Han Chinese. So the Liao, Jin, Yuan or the Qing and some of the earlier dynasties which were actually not founded by Chinese and are often seen as conquest dynasties. But in the model of dynasties they are usually integrated into the Chinese history, to construct a clear line of succession. [Sometimes with the specific purpose to legitimize another conquest dynasty as happened under Yuan for the Liao and the Jin]
Speaking about the distinciveness of the different dynasties themselves between each other this is a very broad topic and I am not sure if I can give it enough justice here but I would just highlight some of the big cornerstones here: As mentioned above the territory ruled by the "legitimate" Han-Chinese dynasties differs very much. The Han only ruled about a very different land than the Qing. The largest extension of the realm would be under the Qing. While the Song are usually considered to be Southwards orientated and smaller. This always also contains questions of culture and identity as the deeper South is traditionally seen as a wasteland or periperhy [the Chinese have terms for western, southern and northern barbarians, therefore the further you go into one of the directions the more barbarian it would get according to traditional Chinese worldview]. Even when you are talking about the Qin, as the Qin state, which was located in the West of China, previous to the unification fo Qin Shihuang, it is clear that they were often see by others as well as by later Han dynasty writers as barbaric. Therefore, I want to highlight that even by the time of Qin they might have appeared uncivilized to some of their enemies while the official history and the reasearch of Chinese history still consideres both sides as Han Chinese!
Especially for the post Han period of the three kingdoms until the reunification China under Sui we then have a time of disunity with the branching of into many blossoming local cultures compared to before. This time coincides with the spread of Buddhism in China and is even by professors of the preimperial or lets say classic period seen as a time that is more difficult to research because of the rise of Buddhist vocabulary and thinking in the texts which markes a change. Accordingly, the Tang dynasty which is seen as the golden age of Chinese history and is the dynasty build upon the former time of disunity with foreign influences and local culture, the Tang is usually seen as being multicultural. At this time you would find many different religions and people in the capital of Tang. This does not mean thou that the Tang was unique in that, but is also a value attributed to them by scholarship.
I hope this response is not too much all over the place, I just think it is very difficult to answer such a big question in short.