Question: I have never really understood the connotation of someone calling their domicile a *condo* over the word *apartment*. I have a vague feeling the former is fancier and more up-scale, but are there any real differences I'm not aware of?

Answer: 
The question as asked assumes an established relationship between "condo" and apartment" that is faulty. This is partly the foundation for the confusion. Even after a lengthy discussion like that above still leaves people confused is because the "definitions" being offered are still unsatisfactory and not conforming to a reality people innately understand but can't articulate.

A condominium (or "condo") is an ownership structure along with "cooperative" and "fee simple". A condo tells you, among other things, how ownership of the unit (residential, retail, commercial, etc,) and common spaces are divided or held. It does not describe any occupancy or physical characteristic of a unit. Any occupancy type (housing, retail, office commercial, industrial, etc.) can be structured as any one of these ownership structures, including as a condominium. 

Therefore, and contrary to common belief, a condominium is not a type of housing. In NYC , Chicago, and many large cities, there are buildings that contain condos but have no residential units. They are office buildings, retail malls, or some combination of the two. The reason why so many people falsely believe that a condominium is a type of housing is because residential condominiums are the only iteration of a condominium with which they are familiar.

An apartment is a descriptive term. It describes a physical characteristic of a residential unit- being that it is one dwelling unit contained in one building structure. Contrary to common belief, an apartment is not housing that is for rent. It may be but not necessarily. Most people come to associate apartments as strictly rental housing because this is the iteration with which they are most familiar. (If apartments were strictly housing for rent, then the phrase "apartment for rent" would be grammatically redundant. It is not.) 

Therefore, an apartment (a residential unit, supposedly among many, in a structure) can itself be structured as a condominium or cooperative. A series of apartments contained in one structure (for example an apartment building) can be structured as fee simple (a building owned by a single entity). The same series of apartments can also be structured as individual condos/coops (owned by several or only one entity). 

You can overlay any ownership structure with any housing description. This can be expanded to whole developments. For example Kinzie Park in Chicago is a 6.5-acre development that has a high-rise tower, a mid-rise building, and many townhouses. All 300 units are structured as condominiums. Even the parking spaces are deeded as condos. You can even have multiple ownership structures in one development with one or multiple occupancies. However most people are already confused enough so I won't add to it.
Condos tend to be in complexes rather than single buildings. Additionally, a condo is purchased and owned, whereas an apartment is rented.

[This Link](http://www.realtor.com/Basics/AllAbout/HomeAffordability/Condo.asp) has some good info about condos.