We tend to think of the nineteenth century as the era in which sail gave way to mechanised forms of shipping (such as steam). But did the century see significant advances in sailing technology as well? What difference, if any, was there between an average sailing ship in 1800 and 1900?

There were many advances inc.

Composite hulls = iron and latterly steel frameworks + wooden hulls / planking. Initially prompted by the shortage of compass timber - which replaced by iron knees. But gradually extended to a full framework, that significantly stronger than all wood ships and notably reduced hogging etc.
The gradual introduction of more durable wire rigging and steel masts.
Somewhat late - just coming into widespread service among merchantmen late 19th. cent. was the adaptation of steam to sail ships e.g. auxiliary engines - whose utility / importance self evident and donkey engines - enabling larger sails, rigging etc. worked by smaller crews.

What can also not be over emphasized was the knock on effects of the steam takeover inc.

The development of ports and introduction of steam tugs pushed and enabled sail merchantmen towards the more extreme streamlined types e.g. clippers. Which no longer had to be generalist e.g. ground to unload and able to be guided out of and into ports.
Notably the earlier steam ships very coal hungry and spawned several times larger fleets of feeder sail colliers. Also carving niches in areas where coal seen as a disadvantage e.g. contaminating tea etc. As this traffic declined, this ready made fleet / system moving to time unimportant, bulk trades - where the steamships could still not compete e.g. grain, hides, oil, wool etc. That still sail dominated 1900.