Here's the real germ of the Cubs/White Sox thing, in my opinion. First, the geography of the city lends itself to to enmity- the North Side really is more affluent, cosmopolitan, etc. The mutual recognition of this fact breeds contempt. Secondly, the Cubs have a far wider fan base- thanks to a tradition (of losing), an old-school well kept ballpark, and just a certain pop culture cache- Harry Caray (who did Sox announcing before Cubs), etc.
When they were owned by the Tribune Co. it was the worst. I remember in '05 when the Sox won the Series the Trib put out a nice little booklet commemorating the season. What's grating is that the Cubs get one of those everytime they make the postseason. I distinctly remember one with "In Dusty We Trusty" on the cover. Ugh. Particularly noxious is the annual ordeal of the Cubs picking up a Big Name for blasphemous amounts of money and watching this player collapse throughout the season. It actually makes me feel bad for the fans. "There's always next year" is the perpetual refrain.
So the Cubs are more popular, have the better ballpark, a fun neighborhood, and a national fanbase. The White Sox fanbase is basically limited to families from the Southside and their descendents. We have a ballpark (getting better) in a not-so-great area, surrounded by trains, a freeway, and parking lots. We have less money, etc.
Which, of course, it what makes it so great to be a Sox fan. It's local, it's familial, it's tribal and primitive in a way that I don't expect others to really understand. The North Side can have the pretty neighborhoods and quaint bars and frat scene [full disclosure: I actually love going out up north]; we'll keep our Sox. It's in the blood and, at least where my family is from, tied up with complex sociological signifiers such as the Daley family, the Democratic party, and the local Catholic parish. The White Sox are more Chicago than Chicago.