One major indicator of what might be called a "modern" city would have to include progressive environmental issues. Things like energy conservation and sustainability have to be part of the new technologies and policies of a modern city.
It's a little unclear in what context the word "modern" is being used. Ancient cities like Tenochtitlan and Baghdad might not be considered modern in some senses, but they contained irrigation systems and aqueducts and fairly well developed city planning for the time, along with complex feats of engineering and construction. I believe these are good examples of what we would call "modern" elements of cities of that day.
Cities in the modern day centuries have been able to support larger populations, even though most of their food and energy comes from outside the cities themselves. So transportation that is efficient and ties it to other cities and resources is one hallmark of a modern city.
In addition, mass transit is common in most modern cities like New York, Moscow, Tokyo and Paris. Cities have also always been the collection point for music and art, and this was also true in the ancient world. This point held true for finance and banking, and centers of trade and industry, which thrive more naturally in modern cities.
Modern cities imply a modern convenience and that would include travel on buses, subways, and trains; modern conveniences like delivery of products and services to the front door, and modern processes of communication, work and lifestyle.