What is a major structural advantage that eukaryotic cells have over prokaryotic cells? Why is this so important?

Eukaryotes have nuclei, organelles, mobility, and variety.

Nuclei - Prokaryotic cells have all of their DNA floating about in the cell. This is highly inefficient, and also very unorganized. Eukaryotic cells have their DNA compiled within a nucleus, which is very space efficient, since they can store thousands of times more chromosomes.

Organelles - Eukaryotic cells have specific organelles designed to fulfill all their needs. This is, again, very efficient. It also allows for more specialization, as different cells can have certain organelles, depending on if they are necessary or not.

Mobility - Most prokaryotes rely on flagella for transportation. While this works, it is far from the most efficient mode of movement. They also are always moving, which isn't necessarily good. Eukaryotes, on the other hand, have complex structures, allowing them to perform more fine movements, and have greater control over their movement in general.

Variety - Similarly to the organelles, the simple fact that Eukaryotes are far more developed and advanced than Prokaryotes means that they have a much wider application, as seen by the fact that Eukaryotes are found in a huge array of different organisms. More features means more things which you can change, which allows for greater specialization, which, in turn, leads to greater efficiency.