A fascinating question and a great answer. The importance of the Exchange to world history can hardly be overstated. Tobacco, maize ("corn" is an old English word for "grain" in general), tomatoes, potatoes, etc. changed the dietary patterns of the entire world. European crops and animals transplanted to the Americas did the same. The importance of the horse in the colonial and Native American cultures is of utmost importance. Native Americans did not have the wheel until Europeans arrived. I doubt there is any way to overestimate the importance of the Columbian Exchange in terms of an expansion of biological diversity.
On the other hand, as Akannan points out, the exchange of ideas and ideals is equally important. The democratic traditions of the native nations of the Eastern seaboard very much affected the colonists in the southern British colonies, expanding their concept of natural rights nearly as much as Locke and Hobbes. The concepts of written language and the domesticity of white lifestyles gave birth to, among other things, the development of individual land ownership, a written language and legislative assembly in the Cherokee nation. Of course the nonsense of the "noble savage" idea was twisted out of all recognition among philosophers in Europe, but even that did give rise to a sort of idealistic egalitarianism.
There were certainly problems, social and biological in terms of misunderstood ideas and the competition of non-native species, an exchange of diseases for which the natives of the New World had no immunity and social problems such as the growing dependence of poor people in Europe on the potato, which had a great effect on later famines in Europe, particularly the Potato Famine in Ireland. Still, disaster and success tend to go hand in hand with humans in general, and the Columbian Exchange is one of the most important ecological events of modern history.
The Columbian Exchange is extremely important on many levels. The primary sphere of relevance is that it marked one of the first moments where there was a pure exchange of ideas, livestock, food, and goods between the world of "the West" and the "New World." One of the critical elements of the age of exploration is the idea of allowing borders and boundaries to be blurred while being simultaneously discovered and etched. The Columbian Exchange was a pivotal moment in this process because it showed that both the nations of Europe were able to directly interact with these new nations whose identity had become of international relevancy. At the same time, the Columbian Exchange became the critical point where the exercise of power and control had become evident in this process of sharing. From an exercise of potential cultural diversity and acceptance, the Columbian Exchange morphed this dialectic into one of control and domination. As goods and ideas were exchanged to Europe, the recipients absorbed new lionesses for which there was no indigenous cure, exploitation of the land as well as its people, and the start of Imperialism. The Columbian Exchange was one sided to this extent, proving that the age of exploration carried much more potential for disaster than anything else. This becomes one of the critical elements of such a dialectic.