They are the yin and the yang of system complexity. Variety describes the degree to which each part of a system can change, on its own. Variety provides the potential for differentiation and change, which creates complexity. The other dimension of system complexity is connectivity. Connectivity creates the potential for integration, where parts or subsystems influence one another, creating complexity.
Other measures of complexity are sometimes used, such as numerousness (the number of parts), but the number of parts is not necessarily a mark of complexity if these parts are just replicated with limited interactions with one another. There are interesting discussions in linguistics for instance, about the complexity of language and its implications on society.
Measures of complexity describe the behavior of systems, rather than their static structure. We can think of many examples where this is true: The World Economy, for instance, depends on each country’s economy, which will change based on local policies and circumstances. The economy of each country is linked to others through global trade. If one country suffers from economic flu, it is likely that others will be affected too. Systems with low variety and low connectivity are simple. Those with high variety and connectivity are complex. This is essentialy a networked view of the World, describing complex systems in terms of the behavior of their nodes (parts, subsystems) and links (connectivity, integration).
Variety and connectivity. Why is this important? This conceptual construct helps us approach complexity and prepare ourselves to deal with it.