I like the explanations given here for the 'Great Cities Dichotomy'. I thought I'd make a handy list of places with the dichotomy and places without and see what it might reveal.
Interestingly, the dichotomy seems to be limited to just North America, Scotland, Australia, New Zealand, South Africa, India, Pakistan, Israel, and Nigeria - all places with former British colonial rule! (And the exceptions which prove the rule: Brazil and Turkey)
Places with the Great Cities Dichotomy:
USA as a whole - Washington, DC versus New York City and LA
Canada as a whole - Ottawa versus Toronto
Scotland - Edinburgh versus Glasgow
Australia - Canberra versus Sydney
New Zealand - Auckland versus Wellington
South Africa - Pretoria versus Johannesburg
India - New Delhi versus Bombay
Pakistan - Islamabad versus Karachi
Israel - Jerusalem versus Tel Aviv
Nigeria - Abuja versus Lagos
Brazil - Brasilia versus Sao Paulo and Rio
Turkey - Ankara versus Istanbul
States of the USA:
Alabama - Montgomery versus Birmingham
Alaska - Juneau versus Anchorage
California - Sacramento versus LA, San Francisco, etc…
Delaware - Dover versus Wilmington
Florida - Tallahassee versus Miami
Illinois - Springfield versus Chicago
Kansas - Topeka versus Wichita
Kentucky - Frankfort versus Louisville
Louisiana - Baton Rouge versus New Orleans
Maine - Augusta versus Portland
Maryland - Annapolis versus Baltimore
Michigan - Lansing versus Detroit
Missouri - Jefferson City versus St. Louis and Kansas City
Nebraska - Lincoln versus Omaha
Nevada - Carson City versus Las Vegas
New Hampshire - Concord versus suburban Boston
New Jersey - Trenton versus NYC suburbia
New Mexico - Santa Fe versus Albuquerque
New York - Albany versus New York City
Ohio - Columbus versus Cleveland and Cincinnati
Oregon - Salem versus Portland
Pennsylvania - Harrisburg versus Philadelphia and Pittsburgh
South Dakota - Pierre versus Souix Falls
Texas - Austin versus Houston and Dallas
Vermont - Montpelier versus Burlington
Virginia - Richmond versus suburban DC and Hampton Area
Washington - Olympia versus Seattle
Wisconsin - Madison versus Milwaukee
Provices of Canada:
Alberta - Edmonton versus Calgary
British Columbia - Victoria versus Vancouver
New Brunswick - Fredrickton versus St. John
Quebec - Quebec City versus Montreal
Saskatchewan - Regina versus Saskatoon
Places WITHOUT the Great Cities Dichotomy:
States of the USA:
Arizona - Phoenix
Arkansas - Little Rock
Colorado - Denver
Connecticut - Hartford
Georgia - Atlanta
Hawaii - Honolulu
Idaho - Boise
Indiana - Indianapolis
Iowa - Des Moines
Massachusetts - Boston
Minnesota - Twin Cities
Mississippi - Jackson
Montana - Helena
North Dakota - Fargo
Oklahoma - Oklahoma City
Rhode Island - Providence
South Carolina - Columbia
Tennessee - Nashville
Utah - Salt Lake City
West Virginia - Charleston
Wyoming - Cheyenne
Provices of Canada:
Manitoba - Winnipeg
Ontario - Toronto
Newfoundland - St. John's
Nova Scotia - Halifax
P.E.I. - Charlottetown
So we see that, for what its worth, roughly equal numbers of states (and provinces) have the dichotomy as don't have it, and there doesn't seem to be any regional patterning. It seems that a more populous state is more likely to have a great cities dichotomy.