much higher-quality steel

Quite the opposite, actually--it is pretty unlikely that the Romans had any real consistent production of artificial steel, while the Han Dynasty not only did but also had cast iron, which would not become available in the west until the early modern period. But generally trying to put two sets of tally marks for "technology" against each other will not yield a particularly informative comparison.

There is a generally plausible argument that the Roman empire was significantly wealthier than the Han empire and had a more complex economy, but I generally feel that until much more archaeological work is done in China that judgement will be a touch dubious. A lot of the transformation of understanding of the Roman economy very the past fifty odd years has been driven by extensive archaeological work, while the archaeology of China is still relatively undeveloped and heavily focused on tombs.

Edit: forgot the source, but Schiedel et al Rome and China: Comparative Perspectives on Ancient World Empires is an excellent work that deals with this topic.

First off, I'll dog pile on here a bit and challenge the idea that you can directly compare the level of "scientific advancement" in two complex, wealthy societies before the advent of anything approaching actual science. Historians really don't like the idea of "technology" as a list of inventions that a civilization or proto-state slowly advances along. It really doesn't work that way. Much of what you might consider "technology" either isn't practically useful in any way to the general society (kites, roman steam engines), or is more an expression of available manpower and resources than anything scientific.

Second, I would like to push back strongly on the idea that Romans had more advanced hydraulic engineering than the Chinese. I'm not as familiar with most of your other examples, but China unquestionably had the most sophisticated understanding and practical implementation of large scale hydraulic engineering of any pre-modern area other than possibly parts of the Netherlands.

China's systems of irrigation, canals, flooding controls, sea walls/reclaimed land, locks, levees, water pumps, polders, were more sophisticated and organized than any Roman systems. This led to a density of agricultural production and a level of direct state control that was not possible in Rome.

Look at things like the 246 BCE Zheng Guo canal. A massive feat of organized civil engineering, it allowed deliberate controlled flooding of farmland with silt-rich, fertilizing water. It also facilitated grain transport and checked soil salinity. This was not just a big ditch - the canal was the product of a very sophisticated understanding of hydrology and relied on a number of technologies. From the rock baskets used to divert water, to roofed areas that protected the canal from gully runoff, to the surveying techniques used, to grilles used to block debris but not water from entering, to the understanding of large scale hydrology and so on.

The Romans had nothing of this scale and sophistication - their largest canals were usually just a few miles long and were usually just used to drain an estuary or marsh or for transport. It's not even the largest ancient Chinese canal - it's dwarfed by the monumental Grand Canal and its precursors.

Stuff like that is harder to see as "technology" than a useless toy like a proto-steam engine, or in the case of hydraulic engineering something like a complicated decorative fountain. But I don't think that's a particularly useful way of looking at the subject.

It's also very important to point out that none of what I said really points to Roman technology being more or less advanced than Chinese, even in the area of hydraulic engineering. Roman technology met Roman needs quite well. Chinese technology met Chinese needs. Because the goals and challenges were not the same, it's very difficult to look at them from the perspective of one "winning" a tech race. Rome did not have the need to connect a series of east to west rivers in a north to south empire the way the Chinese did, nor did their tributary/vassalage/trade system of imperial control immediately reward the intensification of agriculture in the way the more centrally controlled tax based Chinese system did.

Most of this is drawn from Retreat of the Elephants: an Environmental History of China by Mark Elvin. Some comparisons are drawns from Rome and China: Comparative Perspectives on Ancient World Empires By Walter Scheidel.

When people discuss China under the Han Dynasty's rule the main technological feats they mention are [...] windmills and gunpowder (2 things which were only discovered much later, with Rome actually having very primitive windmills before the Chinese)

Although I'm not a historian, I do work as a miller on a historic windmill and have delved quite deep into the history of (wind)mills. What you're writing here is a bit misleading and perhaps only half-true at best.

Although China is sometimes credited with the invention of the windmill, there is no historical or archaeological evidence to support this claim. The earliest mention of a windmill that we know of dates back to the 7th century in Persia, although the authenticity is being questioned because no documents from that age survive. The first certain mention dates to the 10th century in Persia, only about 100 years before the first certain mills in western Europe. These mills had a vertical wind shaft and worked by sheltering one half of the rotor from the wind.

From there the vertical windmill likely spread to India and China, where it is mentioned in the 13th century AD. These Chinese windmills were an evolution of the Persian design and didn't require a windshield. These so-called panemone windmills were used to irrigate ricefields. Picture of a reconstruction

Europe however is a different matter. Water power has been used since antiquity to drive small mills with a horizontal water wheel.

In the first century BC the vertical watermill appeared. This type was described in detail by Roman engineer Vitruvius in 20 BC, who noted its angular transmission to drive the stones: revolutionary at the time and vital to the later windmill. Where the horizontal mill was limited to fast currents, this mill could work in slow rivers and had the potential to provide more power.

But after Vitruvius, despite its clear superiority to the horizontal mill, the vertical watermill seems to be forgotten. A reason for this could be the ample availability of slave labor for manual mills. The mill was again mentioned in the 5th century when slavery was abolished, after that there was a long silence in the dark ages. The watermills that were described during the dark ages were of such low yield they could only be of the horizontal type.

The vertical watermill reappeared in the 9th century in Flanders and from there spread rapidly over western Europe. Around the 11th century nearly all available power from streams and rivers was used for watermills in flat area's so other sources of power were sought.

This logically resulted in the first windmills, which are somewhat of a mystery. Depending on what historian you believe it is apparently invented somewhere in Flanders, Coastal France or England. The first hard textual evidence we have of windmills date from 1180 in Normandy, 1181 in England and 1191 in Flanders. All these texts mention already existing windmills.

What these windmills looked like is a matter of debate. The majority of historians agree they must be primitive versions of the post mill that still exists as a type today. The main problem with the wind in Western Europe is that it blows from different directions. The medieval solution to this was to place the mill on a central post on which it could be turned into the wind. There are few historians and millers who believe the early postmill was essentially a vertical watermill placed on a pole with the wheel swapped for sails. Although it would be a logical progression, there is no evidence for it save a 17th century sketch.

Originally these mills were quite small, and the post and crossplates were buried in the ground to prevent them from being blown over. The remains of buried crossplates have been found all over England and Flanders This subtype has the name 'sunk postmill'. Over the course of centuries this mill evolved -including a few critical changes improving its stability- into the full scale postmill in the mid-15th century that is completely above the ground. As a miller I'm still impressed with the amount of craftsmanship people had to build such a structure. Despite its rickety appearance, this final form of postmill is actually very stable. Many of them have survived for over 400 years.