The longest lasting impact of the World's Columbian Exposition in Chicago is that it was the first large scale application of the City Beautiful Movement, in the United States. It impacted urban planning in many cities during the next two or three decades. It resulted in long lasting features, like the Oakland cultural district in Pittsburgh PA and Monument Avenue in Richmond VA, to give just two examples. The vestiges of the City Beautiful movement can be seen in cities, where monumental Beaux Arts architecture is found along wide avenues that link major parks with the city center.
That is just one impact, that still leaves traces in 2015. There were other cultural impacts that were either not as long lasting, or of less monumental importance. There is a very good book that delves deeply into this World's Fair and its impact on the United States in the twentieth century. "The Devil in the White City: Murder, Magic and Madness at the fair that changed America" by Eric Larson. This book is a must read if you have a great interest in the 1893 Columbia Exposition of Chicago.